{"id":90835,"date":"2026-08-06T05:48:48","date_gmt":"2026-08-05T21:48:48","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/90835.html"},"modified":"2026-08-06T05:48:48","modified_gmt":"2026-08-05T21:48:48","slug":"naive-bayes%e3%80%81%e6%9c%80%e5%a4%a7%e4%bc%bc%e7%84%b6%e4%bc%b0%e8%ae%a1%e4%b8%8e%e6%9c%80%e5%a4%a7%e5%90%8e%e9%aa%8c%e4%bc%b0%e8%ae%a1","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/90835.html","title":{"rendered":"Naive Bayes\u3001\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0e\u6700\u5927\u540e\u9a8c\u4f30\u8ba1"},"content":{"rendered":"<h2>Naive Bayes\u3001\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0e\u6700\u5927\u540e\u9a8c\u4f30\u8ba1<\/h2>\n<p>\u672c\u6587\u7cfb\u7edf\u4ecb\u7ecd\u4e09\u4e2a\u7d27\u5bc6\u76f8\u5173\u7684\u6982\u7387\u7edf\u8ba1\u6982\u5ff5&#xff1a;Naive Bayes\u3001Maximum Likelihood Estimation&#xff08;ML&#xff09;\u4e0e Maximum A Posteriori Estimation&#xff08;MAP&#xff09;\u3002\u5b83\u4eec\u4e0d\u4ec5\u662f\u4f20\u7edf\u673a\u5668\u5b66\u4e60\u7684\u91cd\u8981\u57fa\u7840&#xff0c;\u4e5f\u6784\u6210\u4e86\u7406\u89e3\u6982\u7387\u6a21\u578b\u3001\u8d1d\u53f6\u65af\u5b66\u4e60\u3001\u6b63\u5219\u5316\u548c\u6df1\u5ea6\u5b66\u4e60\u8bad\u7ec3\u76ee\u6807\u7684\u91cd\u8981\u5165\u53e3\u3002<\/p>\n<hr \/>\n<h3>\u4e00\u3001\u4e3a\u4ec0\u4e48\u8981\u540c\u65f6\u5b66\u4e60 Naive Bayes\u3001ML \u548c MAP<\/h3>\n<p>\u8fd9\u4e09\u4e2a\u6982\u5ff5\u5904\u5728\u540c\u4e00\u6761\u903b\u8f91\u94fe\u4e0a&#xff1a;<\/p>\n<li>Naive Bayes \u662f\u4e00\u79cd\u57fa\u4e8e\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u5206\u7c7b\u6a21\u578b\u3002<\/li>\n<li>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1 ML \u7528\u8bad\u7ec3\u6570\u636e\u5bfb\u627e\u6700\u53ef\u80fd\u7684\u6a21\u578b\u53c2\u6570\u3002<\/li>\n<li>\u6700\u5927\u540e\u9a8c\u4f30\u8ba1 MAP \u5728 ML \u7684\u57fa\u7840\u4e0a\u52a0\u5165\u53c2\u6570\u5148\u9a8c\u3002<\/li>\n<li>\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u4ea4\u53c9\u71b5\u8bad\u7ec3\u3001\u6743\u91cd\u8870\u51cf\u548c\u8d1d\u53f6\u65af\u795e\u7ecf\u7f51\u7edc&#xff0c;\u90fd\u53ef\u4ee5\u4ece\u8fd9\u4e09\u4e2a\u6982\u5ff5\u4e2d\u627e\u5230\u7406\u8bba\u6765\u6e90\u3002<\/li>\n<p>\u53ef\u4ee5\u5148\u7528\u4e00\u53e5\u8bdd\u533a\u5206&#xff1a;<\/p>\n<ul>\n<li>ML&#xff1a;\u53ea\u770b\u6570\u636e&#xff0c;\u5bfb\u627e\u6700\u80fd\u89e3\u91ca\u6570\u636e\u7684\u53c2\u6570\u3002<\/li>\n<li>MAP&#xff1a;\u540c\u65f6\u770b\u6570\u636e\u548c\u5148\u9a8c&#xff0c;\u5bfb\u627e\u540e\u9a8c\u6982\u7387\u6700\u5927\u7684\u53c2\u6570\u3002<\/li>\n<li>Naive Bayes&#xff1a;\u5229\u7528\u7c7b\u522b\u5148\u9a8c\u548c\u7279\u5f81\u6761\u4ef6\u6982\u7387\u5b8c\u6210\u5206\u7c7b\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u4e8c\u3001\u8d1d\u53f6\u65af\u5b9a\u7406<\/h3>\n<p>\u8d1d\u53f6\u65af\u5b9a\u7406\u63cf\u8ff0\u4e86\u5728\u89c2\u5bdf\u5230\u8bc1\u636e\u4e4b\u540e&#xff0c;\u5982\u4f55\u66f4\u65b0\u5bf9\u67d0\u4e2a\u4e8b\u4ef6\u6216\u53c2\u6570\u7684\u8ba4\u8bc6\u3002<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(A\u2223B)&#061;P(B\u2223A)P(A)P(B)P(A|B)&#061;\\\\frac{P(B|A)P(A)}{P(B)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0502em\">B<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>P(A)&#xff1a;\u5148\u9a8c\u6982\u7387&#xff0c;\u5373\u89c2\u5bdf\u8bc1\u636e\u524d\u5bf9\u4e8b\u4ef6 A \u7684\u8ba4\u8bc6\u3002<\/li>\n<li>P(B|A)&#xff1a;\u4f3c\u7136&#xff0c;\u5373\u4e8b\u4ef6 A \u6210\u7acb\u65f6\u89c2\u5bdf\u5230 B \u7684\u6982\u7387\u3002<\/li>\n<li>P(B)&#xff1a;\u8bc1\u636e\u6982\u7387&#xff0c;\u4e5f\u53eb\u8fb9\u7f18\u4f3c\u7136\u3002<\/li>\n<li>P(A|B)&#xff1a;\u540e\u9a8c\u6982\u7387&#xff0c;\u5373\u89c2\u5bdf\u5230 B \u540e\u4e8b\u4ef6 A \u7684\u6982\u7387\u3002<\/li>\n<\/ul>\n<p>\u5728\u53c2\u6570\u4f30\u8ba1\u95ee\u9898\u4e2d&#xff0c;\u901a\u5e38\u5199\u6210&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(\u03b8\u2223D)&#061;P(D\u2223\u03b8)P(\u03b8)P(D)P(\\\\theta|D)&#061;\\\\frac{P(D|\\\\theta)P(\\\\theta)}{P(D)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\u03b8 \u8868\u793a\u6a21\u578b\u53c2\u6570\u3002<\/li>\n<li>D \u8868\u793a\u8bad\u7ec3\u6570\u636e\u3002<\/li>\n<li>P(\u03b8) \u662f\u53c2\u6570\u5148\u9a8c\u3002<\/li>\n<li>P(D|\u03b8) \u662f\u4f3c\u7136\u51fd\u6570\u3002<\/li>\n<li>P(\u03b8|D) \u662f\u53c2\u6570\u540e\u9a8c\u3002<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260805214844-6a73afbc7e542.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u8d1d\u53f6\u65af\u5b66\u4e60\u7684\u6838\u5fc3\u53ef\u4ee5\u6982\u62ec\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">Posterior\u221dLikelihood\u00d7Prior\\\\text{Posterior}\\\\propto\\\\text{Likelihood}\\\\times\\\\text{Prior}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord text\"><span class=\"mord\">Posterior<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u221d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em;vertical-align: -0.0833em\"><\/span><span class=\"mord text\"><span class=\"mord\">Likelihood<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord text\"><span class=\"mord\">Prior<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u4e3a\u5bf9\u4e8e\u56fa\u5b9a\u6570\u636e\u96c6 D&#xff0c;\u8bc1\u636e P(D) \u4e0e\u53c2\u6570 \u03b8 \u65e0\u5173&#xff0c;\u6240\u4ee5\u5728\u4f18\u5316\u53c2\u6570\u65f6\u901a\u5e38\u53ef\u4ee5\u5ffd\u7565\u5b83\u3002<\/p>\n<hr \/>\n<h3>\u4e09\u3001Maximum Likelihood Estimation<\/h3>\n<h4>3.1 \u4ec0\u4e48\u662f\u6700\u5927\u4f3c\u7136\u4f30\u8ba1<\/h4>\n<p>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u7684\u76ee\u6807\u662f&#xff1a;<\/p>\n<p>\u627e\u5230\u4e00\u7ec4\u53c2\u6570&#xff0c;\u4f7f\u5df2\u7ecf\u89c2\u5bdf\u5230\u7684\u8bad\u7ec3\u6570\u636e\u51fa\u73b0\u7684\u6982\u7387\u6700\u5927\u3002<\/p>\n<p>\u8bbe\u8bad\u7ec3\u6570\u636e\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">D&#061;{x1,x2,\u2026,xN}D&#061;\\\\{x_1,x_2,\\\\ldots,x_N\\\\}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6a21\u578b\u53c2\u6570\u4e3a \u03b8\u3002\u5982\u679c\u6837\u672c\u72ec\u7acb\u540c\u5206\u5e03&#xff0c;\u5219\u6574\u4e2a\u6570\u636e\u96c6\u7684\u4f3c\u7136\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(D\u2223\u03b8)&#061;\u220fi&#061;1NP(xi\u2223\u03b8)P(D|\\\\theta)&#061;\\\\prod_{i&#061;1}^{N}P(x_i|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^ML&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)\\\\hat{\\\\theta}_{ML}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7531\u4e8e\u8fde\u4e58\u5bb9\u6613\u9020\u6210\u6570\u503c\u4e0b\u6ea2&#xff0c;\u800c\u4e14\u5bf9\u6570\u51fd\u6570\u662f\u5355\u8c03\u9012\u589e\u51fd\u6570&#xff0c;\u6240\u4ee5\u901a\u5e38\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^ML&#061;arg\u2061max\u2061\u03b8log\u2061P(D\u2223\u03b8)\\\\hat{\\\\theta}_{ML}&#061;\\\\arg\\\\max_{\\\\theta}\\\\log P(D|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u72ec\u7acb\u6837\u672c\u4e0b&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">log\u2061P(D\u2223\u03b8)&#061;\u2211i&#061;1Nlog\u2061P(xi\u2223\u03b8)\\\\log P(D|\\\\theta)&#061;\\\\sum_{i&#061;1}^{N}\\\\log P(x_i|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u6b64&#xff0c;\u4e58\u6cd5\u88ab\u8f6c\u6362\u4e3a\u52a0\u6cd5&#xff0c;\u8ba1\u7b97\u66f4\u52a0\u7a33\u5b9a\u3002<\/p>\n<hr \/>\n<h4>3.2 \u9ad8\u65af\u5206\u5e03\u5747\u503c\u7684 ML \u4f30\u8ba1<\/h4>\n<p>\u5047\u8bbe\u6837\u672c\u6765\u81ea\u5747\u503c\u4e3a \u03bc\u3001\u65b9\u5dee\u4e3a\u5df2\u77e5\u5e38\u6570 \u03c3\u00b2 \u7684\u9ad8\u65af\u5206\u5e03&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">xi\u223cN(\u03bc,\u03c32)x_i\\\\sim\\\\mathcal{N}(\\\\mu,\\\\sigma^2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u223c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1474em\">N<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9ad8\u65af\u5206\u5e03\u5bc6\u5ea6\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xi\u2223\u03bc)&#061;12\u03c0\u03c32exp\u2061(\u2212(xi\u2212\u03bc)22\u03c32)P(x_i|\\\\mu)&#061;\\\\frac{1}{\\\\sqrt{2\\\\pi\\\\sigma^2}}\\\\exp\\\\left(-\\\\frac{(x_i-\\\\mu)^2}{2\\\\sigma^2}\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4411em;vertical-align: -0.95em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.1549em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9551em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -2.9151em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0849em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.93em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">exp<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6240\u6709\u6837\u672c\u7684\u5bf9\u6570\u4f3c\u7136\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">log\u2061P(D\u2223\u03bc)&#061;\u2212N2log\u2061(2\u03c0\u03c32)\u221212\u03c32\u2211i&#061;1N(xi\u2212\u03bc)2\\\\log P(D|\\\\mu)&#061;-\\\\frac{N}{2}\\\\log(2\\\\pi\\\\sigma^2)-\\\\frac{1}{2\\\\sigma^2}\\\\sum_{i&#061;1}^{N}(x_i-\\\\mu)^2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0463em;vertical-align: -0.686em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u524d\u534a\u90e8\u5206\u4e0e \u03bc \u65e0\u5173&#xff0c;\u56e0\u6b64\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u7b49\u4ef7\u4e8e\u6700\u5c0f\u5316\u5e73\u65b9\u8bef\u5dee&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bc^ML&#061;arg\u2061min\u2061\u03bc\u2211i&#061;1N(xi\u2212\u03bc)2\\\\hat{\\\\mu}_{ML}&#061;\\\\arg\\\\min_{\\\\mu}\\\\sum_{i&#061;1}^{N}(x_i-\\\\mu)^2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.4em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">\u03bc<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8361em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u7ec8\u5f97\u5230&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bc^ML&#061;1N\u2211i&#061;1Nxi\\\\hat{\\\\mu}_{ML}&#061;\\\\frac{1}{N}\\\\sum_{i&#061;1}^{N}x_i<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u6b63\u6001\u5206\u5e03\u5747\u503c\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u5c31\u662f\u6837\u672c\u5747\u503c\u3002<\/p>\n<hr \/>\n<h4>3.3 ML \u4e0e\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u635f\u5931\u51fd\u6570<\/h4>\n<p>\u6df1\u5ea6\u5b66\u4e60\u8bad\u7ec3\u901a\u5e38\u53ef\u4ee5\u89e3\u91ca\u4e3a\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u3002<\/p>\n<p>\u5bf9\u4e8e\u5206\u7c7b\u95ee\u9898&#xff0c;\u795e\u7ecf\u7f51\u7edc\u8f93\u51fa\u7c7b\u522b\u6982\u7387&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(y&#061;k\u2223x;\u03b8)P(y&#061;k|x;\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6574\u4e2a\u6570\u636e\u96c6\u7684\u6761\u4ef6\u4f3c\u7136\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(D\u2223\u03b8)&#061;\u220fi&#061;1NP(yi\u2223xi;\u03b8)P(D|\\\\theta)&#061;\\\\prod_{i&#061;1}^{N}P(y_i|x_i;\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u7b49\u4ef7\u4e8e\u6700\u5c0f\u5316\u8d1f\u5bf9\u6570\u4f3c\u7136&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">LNLL(\u03b8)&#061;\u2212\u2211i&#061;1Nlog\u2061P(yi\u2223xi;\u03b8)\\\\mathcal{L}_{NLL}(\\\\theta)&#061;-\\\\sum_{i&#061;1}^{N}\\\\log P(y_i|x_i;\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathcal\">L<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mord mathnormal mtight\">LL<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5bf9\u4e8e one-hot \u6807\u7b7e\u548c Softmax \u8f93\u51fa&#xff0c;\u8d1f\u5bf9\u6570\u4f3c\u7136\u5c31\u662f\u4ea4\u53c9\u71b5\u635f\u5931&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">LCE&#061;\u2212\u2211i&#061;1N\u2211k&#061;1Kyiklog\u2061y^ik\\\\mathcal{L}_{CE}&#061;-\\\\sum_{i&#061;1}^{N}\\\\sum_{k&#061;1}^{K}y_{ik}\\\\log\\\\hat{y}_{ik}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathcal\">L<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0576em\">E<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.1304em;vertical-align: -1.3021em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3021em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">ik<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">ik<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u6b64&#xff0c;\u53ef\u4ee5\u628a\u666e\u901a\u795e\u7ecf\u7f51\u7edc\u5206\u7c7b\u8bad\u7ec3\u7406\u89e3\u4e3a&#xff1a;<\/p>\n<p>\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6c42\u89e3\u6a21\u578b\u53c2\u6570\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u3002<\/p>\n<hr \/>\n<h3>\u56db\u3001Maximum A Posteriori Estimation<\/h3>\n<h4>4.1 \u4ec0\u4e48\u662f\u6700\u5927\u540e\u9a8c\u4f30\u8ba1<\/h4>\n<p>\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u5bfb\u627e\u4f7f\u540e\u9a8c\u6982\u7387\u6700\u5927\u7684\u53c2\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061max\u2061\u03b8P(\u03b8\u2223D)\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\max_{\\\\theta}P(\\\\theta|D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6839\u636e\u8d1d\u53f6\u65af\u5b9a\u7406&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(\u03b8\u2223D)&#061;P(D\u2223\u03b8)P(\u03b8)P(D)P(\\\\theta|D)&#061;\\\\frac{P(D|\\\\theta)P(\\\\theta)}{P(D)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7531\u4e8e P(D) \u4e0d\u968f \u03b8 \u53d8\u5316&#xff0c;\u56e0\u6b64&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)P(\u03b8)\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)P(\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u53d6\u5bf9\u6570\u540e&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061max\u2061\u03b8[log\u2061P(D\u2223\u03b8)&#043;log\u2061P(\u03b8)]\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\max_{\\\\theta}\\\\left[\\\\log P(D|\\\\theta)&#043;\\\\log P(\\\\theta)\\\\right]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">[<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e5f\u53ef\u4ee5\u5199\u6210\u6700\u5c0f\u5316\u5f62\u5f0f&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061min\u2061\u03b8[\u2212log\u2061P(D\u2223\u03b8)\u2212log\u2061P(\u03b8)]\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\min_{\\\\theta}\\\\left[-\\\\log P(D|\\\\theta)-\\\\log P(\\\\theta)\\\\right]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">[<\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u8bf4\u660e MAP \u7684\u76ee\u6807\u7531\u4e24\u90e8\u5206\u6784\u6210&#xff1a;<\/p>\n<li>\u6570\u636e\u635f\u5931&#xff1a;\u8981\u6c42\u53c2\u6570\u80fd\u591f\u89e3\u91ca\u8bad\u7ec3\u6570\u636e\u3002<\/li>\n<li>\u5148\u9a8c\u7ea6\u675f&#xff1a;\u8981\u6c42\u53c2\u6570\u7b26\u5408\u6211\u4eec\u9884\u5148\u8bbe\u5b9a\u7684\u5408\u7406\u8303\u56f4\u3002<\/li>\n<hr \/>\n<h4>4.2 ML \u4e0e MAP \u7684\u56fe\u5f62\u5bf9\u6bd4<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260805214845-6a73afbd81fd7.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u56fe\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\u4f3c\u7136\u66f2\u7ebf\u7684\u6700\u9ad8\u70b9\u5bf9\u5e94 ML \u4f30\u8ba1\u3002<\/li>\n<li>\u5148\u9a8c\u8868\u8fbe\u4e86\u53c2\u6570\u5728\u89c2\u5bdf\u6570\u636e\u524d\u7684\u504f\u597d\u3002<\/li>\n<li>\u540e\u9a8c\u7531\u4f3c\u7136\u4e0e\u5148\u9a8c\u5171\u540c\u51b3\u5b9a\u3002<\/li>\n<li>\u540e\u9a8c\u6700\u9ad8\u70b9\u5bf9\u5e94 MAP \u4f30\u8ba1\u3002<\/li>\n<\/ul>\n<p>\u5f53\u6570\u636e\u91cf\u8f83\u5c11\u65f6&#xff0c;\u5148\u9a8c\u5bf9 MAP \u7684\u5f71\u54cd\u8f83\u5927\u3002<\/p>\n<p>\u5f53\u6570\u636e\u91cf\u975e\u5e38\u5927\u65f6&#xff0c;\u4f3c\u7136\u901a\u5e38\u5360\u4e3b\u5bfc\u5730\u4f4d&#xff0c;MAP \u4e0e ML \u4f1a\u9010\u6e10\u63a5\u8fd1\u3002<\/p>\n<hr \/>\n<h4>4.3 \u9ad8\u65af\u5148\u9a8c\u4e0e L2 \u6b63\u5219\u5316<\/h4>\n<p>\u5047\u8bbe\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\u670d\u4ece\u96f6\u5747\u503c\u9ad8\u65af\u5148\u9a8c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(\u03b8)\u221dexp\u2061(\u2212\u03bb2\u2225\u03b8\u222522)P(\\\\theta)\\\\propto\\\\exp\\\\left(-\\\\frac{\\\\lambda}{2}\\\\|\\\\theta\\\\|_2^2\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u221d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mop\">exp<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bb<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -2.453em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u53d6\u8d1f\u5bf9\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212log\u2061P(\u03b8)&#061;\u03bb2\u2225\u03b8\u222522&#043;C-\\\\log P(\\\\theta)&#061;\\\\frac{\\\\lambda}{2}\\\\|\\\\theta\\\\|_2^2&#043;C<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bb<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -2.453em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4ee3\u5165 MAP \u76ee\u6807&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061min\u2061\u03b8[Ldata(\u03b8)&#043;\u03bb2\u2225\u03b8\u222522]\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\min_{\\\\theta}\\\\left[\\\\mathcal{L}_{data}(\\\\theta)&#043;\\\\frac{\\\\lambda}{2}\\\\|\\\\theta\\\\|_2^2\\\\right]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4em;vertical-align: -0.95em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">[<\/span><\/span><span class=\"mord\"><span class=\"mord mathcal\">L<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bb<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -2.453em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size3\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u6b63\u662f\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684 L2 \u6b63\u5219\u5316\u6216\u6743\u91cd\u8870\u51cf\u5f62\u5f0f\u3002<\/p>\n<p>\u56e0\u6b64&#xff1a;<\/p>\n<p>L2 \u6b63\u5219\u5316\u53ef\u4ee5\u89e3\u91ca\u4e3a\u5bf9\u6a21\u578b\u53c2\u6570\u65bd\u52a0\u96f6\u5747\u503c\u9ad8\u65af\u5148\u9a8c\u540e\u8fdb\u884c MAP \u4f30\u8ba1\u3002<\/p>\n<hr \/>\n<h4>4.4 \u62c9\u666e\u62c9\u65af\u5148\u9a8c\u4e0e L1 \u6b63\u5219\u5316<\/h4>\n<p>\u82e5\u53c2\u6570\u670d\u4ece\u96f6\u5747\u503c\u62c9\u666e\u62c9\u65af\u5206\u5e03&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(\u03b8)\u221dexp\u2061(\u2212\u03bb\u2225\u03b8\u22251)P(\\\\theta)\\\\propto\\\\exp(-\\\\lambda\\\\|\\\\theta\\\\|_1)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u221d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">exp<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">\u03bb<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5219\u8d1f\u5bf9\u6570\u5148\u9a8c\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u2212log\u2061P(\u03b8)&#061;\u03bb\u2225\u03b8\u22251&#043;C-\\\\log P(\\\\theta)&#061;\\\\lambda\\\\|\\\\theta\\\\|_1&#043;C<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">\u03bb<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>MAP \u76ee\u6807\u53d8\u6210&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061min\u2061\u03b8[Ldata(\u03b8)&#043;\u03bb\u2225\u03b8\u22251]\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\min_{\\\\theta}\\\\left[\\\\mathcal{L}_{data}(\\\\theta)&#043;\\\\lambda\\\\|\\\\theta\\\\|_1\\\\right]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">min<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\">[<\/span><span class=\"mord\"><span class=\"mord mathcal\">L<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">\u03bb<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u5c31\u662f L1 \u6b63\u5219\u5316\u3002<\/p>\n<p>L1 \u6b63\u5219\u5316\u503e\u5411\u4e8e\u4ea7\u751f\u7a00\u758f\u53c2\u6570&#xff0c;\u56e0\u6b64\u5e38\u7528\u4e8e\u7279\u5f81\u9009\u62e9\u548c\u6a21\u578b\u538b\u7f29\u3002<\/p>\n<hr \/>\n<h3>\u4e94\u3001ML \u4e0e MAP \u7684\u6838\u5fc3\u533a\u522b<\/h3>\n<table>\n<tr>\u5bf9\u6bd4\u7ef4\u5ea6MLMAP<\/tr>\n<tbody>\n<tr>\n<td>\u4f18\u5316\u76ee\u6807<\/td>\n<td>\u6700\u5927\u5316\u4f3c\u7136<\/td>\n<td>\u6700\u5927\u5316\u540e\u9a8c<\/td>\n<\/tr>\n<tr>\n<td>\u662f\u5426\u4f7f\u7528\u5148\u9a8c<\/td>\n<td>\u5426<\/td>\n<td>\u662f<\/td>\n<\/tr>\n<tr>\n<td>\u6570\u636e\u8f83\u5c11\u65f6<\/td>\n<td>\u5bb9\u6613\u53d7\u566a\u58f0\u5f71\u54cd<\/td>\n<td>\u5148\u9a8c\u53ef\u63d0\u4f9b\u7ea6\u675f<\/td>\n<\/tr>\n<tr>\n<td>\u6570\u636e\u91cf\u5f88\u5927\u65f6<\/td>\n<td>\u901a\u5e38\u7a33\u5b9a<\/td>\n<td>\u5f80\u5f80\u9010\u6e10\u63a5\u8fd1 ML<\/td>\n<\/tr>\n<tr>\n<td>\u4e0e\u6b63\u5219\u5316\u5173\u7cfb<\/td>\n<td>\u901a\u5e38\u65e0\u663e\u5f0f\u5148\u9a8c<\/td>\n<td>\u53ef\u89e3\u91ca\u5e38\u89c1\u6b63\u5219\u5316<\/td>\n<\/tr>\n<tr>\n<td>\u53c2\u6570\u542b\u4e49<\/td>\n<td>\u56fa\u5b9a\u4f46\u672a\u77e5<\/td>\n<td>\u5e26\u6709\u6982\u7387\u5148\u9a8c\u7684\u672a\u77e5\u53c2\u6570<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u4e8c\u8005\u7684\u5173\u7cfb\u53ef\u4ee5\u5199\u6210&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^ML&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)\\\\hat{\\\\theta}_{ML}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)P(\u03b8)\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)P(\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5982\u679c\u53c2\u6570\u5148\u9a8c\u662f\u5747\u5300\u5206\u5e03&#xff0c;\u5219 P(\u03b8) \u5bf9\u6240\u6709\u53c2\u6570\u53d6\u503c\u76f8\u540c&#xff0c;\u6b64\u65f6&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;\u03b8^ML\\\\hat{\\\\theta}_{MAP}&#061;\\\\hat{\\\\theta}_{ML}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<hr \/>\n<h3>\u516d\u3001Naive Bayes<\/h3>\n<h4>6.1 Naive Bayes \u662f\u4ec0\u4e48<\/h4>\n<p>Naive Bayes \u662f\u4e00\u7c7b\u57fa\u4e8e\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u76d1\u7763\u5206\u7c7b\u7b97\u6cd5\u3002<\/p>\n<p>\u5bf9\u4e8e\u8f93\u5165\u7279\u5f81&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">x&#061;(x1,x2,\u2026,xn)x&#061;(x_1,x_2,\\\\ldots,x_n)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u76ee\u6807\u662f\u8ba1\u7b97\u6bcf\u4e2a\u7c7b\u522b C_k \u7684\u540e\u9a8c\u6982\u7387&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(Ck\u2223x)&#061;P(x\u2223Ck)P(Ck)P(x)P(C_k|x)&#061;\\\\frac{P(x|C_k)P(C_k)}{P(x)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u4e3a P(x) \u5bf9\u6240\u6709\u7c7b\u522b\u76f8\u540c&#xff0c;\u6240\u4ee5\u5206\u7c7b\u65f6\u53ea\u9700\u8981\u6bd4\u8f83&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(Ck\u2223x)\u221dP(Ck)P(x\u2223Ck)P(C_k|x)\\\\propto P(C_k)P(x|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u221d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>Naive Bayes \u7684\u201c\u6734\u7d20\u201d\u6765\u81ea\u4e00\u4e2a\u5f3a\u5047\u8bbe&#xff1a;<\/p>\n<p>\u5728\u7ed9\u5b9a\u7c7b\u522b\u7684\u6761\u4ef6\u4e0b&#xff0c;\u5404\u4e2a\u7279\u5f81\u76f8\u4e92\u72ec\u7acb\u3002<\/p>\n<p>\u4e8e\u662f&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(x\u2223Ck)&#061;\u220fj&#061;1nP(xj\u2223Ck)P(x|C_k)&#061;\\\\prod_{j&#061;1}^{n}P(x_j|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0652em;vertical-align: -1.4138em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u7ec8\u5206\u7c7b\u89c4\u5219\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">y^&#061;arg\u2061max\u2061CkP(Ck)\u220fj&#061;1nP(xj\u2223Ck)\\\\hat{y}&#061;\\\\arg\\\\max_{C_k}P(C_k)\\\\prod_{j&#061;1}^{n}P(x_j|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0652em;vertical-align: -1.4138em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3448em\"><span class=\"\" style=\"top: -2.3488em;margin-left: -0.0715em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1512em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8502em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e3a\u4e86\u907f\u514d\u6982\u7387\u8fde\u4e58\u4e0b\u6ea2&#xff0c;\u5b9e\u9645\u8ba1\u7b97\u901a\u5e38\u4f7f\u7528\u5bf9\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">y^&#061;arg\u2061max\u2061Ck[log\u2061P(Ck)&#043;\u2211j&#061;1nlog\u2061P(xj\u2223Ck)]\\\\hat{y}&#061;\\\\arg\\\\max_{C_k}\\\\left[\\\\log P(C_k)&#043;\\\\sum_{j&#061;1}^{n}\\\\log P(x_j|C_k)\\\\right]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.1638em;vertical-align: -1.4138em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3448em\"><span class=\"\" style=\"top: -2.3488em;margin-left: -0.0715em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1512em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8502em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">[<\/span><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/08\/20260805214846-6a73afbe373af.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<hr \/>\n<h4>6.2 \u4e3a\u4ec0\u4e48\u6761\u4ef6\u72ec\u7acb\u5047\u8bbe\u5f88\u91cd\u8981<\/h4>\n<p>\u5982\u679c\u4e0d\u505a\u72ec\u7acb\u6027\u5047\u8bbe&#xff0c;\u9700\u8981\u76f4\u63a5\u4f30\u8ba1&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(x1,x2,\u2026,xn\u2223Ck)P(x_1,x_2,\\\\ldots,x_n|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5f53\u7279\u5f81\u7ef4\u5ea6\u5f88\u9ad8\u65f6&#xff0c;\u8054\u5408\u6982\u7387\u5206\u5e03\u9700\u8981\u5927\u91cf\u6570\u636e\u624d\u80fd\u53ef\u9760\u4f30\u8ba1\u3002<\/p>\n<p>\u6761\u4ef6\u72ec\u7acb\u5047\u8bbe\u5c06\u590d\u6742\u8054\u5408\u6982\u7387\u62c6\u6210\u591a\u4e2a\u4e00\u7ef4\u6761\u4ef6\u6982\u7387&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(x1,x2,\u2026,xn\u2223Ck)&#061;\u220fj&#061;1nP(xj\u2223Ck)P(x_1,x_2,\\\\ldots,x_n|C_k)&#061;\\\\prod_{j&#061;1}^{n}P(x_j|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0652em;vertical-align: -1.4138em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u4f1a\u663e\u8457\u964d\u4f4e\u53c2\u6570\u6570\u91cf\u548c\u8ba1\u7b97\u590d\u6742\u5ea6\u3002<\/p>\n<p>\u867d\u7136\u73b0\u5b9e\u4e2d\u7684\u7279\u5f81\u901a\u5e38\u4e0d\u5b8c\u5168\u72ec\u7acb&#xff0c;\u4f46 Naive Bayes \u5728\u6587\u672c\u5206\u7c7b\u3001\u5783\u573e\u90ae\u4ef6\u8bc6\u522b\u548c\u5c0f\u6837\u672c\u4efb\u52a1\u4e2d\u4ecd\u5e38\u6709\u5f88\u5f3a\u8868\u73b0\u3002<\/p>\n<hr \/>\n<h3>\u4e03\u3001Naive Bayes \u7684\u4e3b\u8981\u7c7b\u578b<\/h3>\n<h4>7.1 Gaussian Naive Bayes<\/h4>\n<p>\u9002\u7528\u4e8e\u8fde\u7eed\u7279\u5f81&#xff0c;\u5e76\u5047\u8bbe\u6bcf\u4e2a\u7c7b\u522b\u4e0b\u7684\u6bcf\u4e2a\u7279\u5f81\u670d\u4ece\u9ad8\u65af\u5206\u5e03&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;12\u03c0\u03c3kj2exp\u2061(\u2212(xj\u2212\u03bckj)22\u03c3kj2)P(x_j|C_k)&#061;\\\\frac{1}{\\\\sqrt{2\\\\pi\\\\sigma_{kj}^2}}\\\\exp\\\\left(-\\\\frac{(x_j-\\\\mu_{kj})^2}{2\\\\sigma_{kj}^2}\\\\right)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.48em;vertical-align: -1.73em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.11em\"><span class=\"pstrut\" style=\"height: 3.1642em\"><\/span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1642em\"><span class=\"svg-align\" style=\"top: -3.8em\"><span class=\"pstrut\" style=\"height: 3.8em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7959em\"><span class=\"\" style=\"top: -2.3987em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.0448em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4374em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.1242em\"><span class=\"pstrut\" style=\"height: 3.8em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.88em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6758em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.3942em\"><span class=\"pstrut\" style=\"height: 3.1642em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.8412em\"><span class=\"pstrut\" style=\"height: 3.1642em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.73em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">exp<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">(<\/span><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7959em\"><span class=\"\" style=\"top: -2.3987em;margin-left: -0.0359em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.0448em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4374em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1234em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8bad\u7ec3\u65f6\u9700\u8981\u4f30\u8ba1&#xff1a;<\/p>\n<ul>\n<li>\u7c7b\u522b\u5148\u9a8c P(C_k)\u3002<\/li>\n<li>\u6bcf\u4e2a\u7c7b\u522b\u4e0b\u6bcf\u4e2a\u7279\u5f81\u7684\u5747\u503c \u03bc_kj\u3002<\/li>\n<li>\u6bcf\u4e2a\u7c7b\u522b\u4e0b\u6bcf\u4e2a\u7279\u5f81\u7684\u65b9\u5dee \u03c3\u00b2_kj\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u4e9b\u53c2\u6570\u901a\u5e38\u4f7f\u7528 ML \u4f30\u8ba1\u3002<\/p>\n<hr \/>\n<h4>7.2 Multinomial Naive Bayes<\/h4>\n<p>\u9002\u7528\u4e8e\u79bb\u6563\u8ba1\u6570\u7279\u5f81&#xff0c;\u5c24\u5176\u9002\u5408\u6587\u672c\u8bcd\u9891\u3002<\/p>\n<p>\u8bbe\u8bcd j \u5728\u7c7b\u522b C_k \u4e2d\u7684\u6761\u4ef6\u6982\u7387\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;\u03b8kjP(x_j|C_k)&#061;\\\\theta_{kj}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e00\u7bc7\u6587\u6863\u4e2d\u5404\u8bcd\u51fa\u73b0\u6b21\u6570\u6784\u6210\u591a\u9879\u5206\u5e03\u3002\u5206\u7c7b\u5f97\u5206\u53ef\u5199\u6210&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">log\u2061P(Ck\u2223x)\u221dlog\u2061P(Ck)&#043;\u2211j&#061;1nxjlog\u2061\u03b8kj\\\\log P(C_k|x)\\\\propto\\\\log P(C_k)&#043;\\\\sum_{j&#061;1}^{n}x_j\\\\log\\\\theta_{kj}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u221d<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0652em;vertical-align: -1.4138em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d x_j \u8868\u793a\u8bcd j \u5728\u6587\u6863\u4e2d\u7684\u51fa\u73b0\u6b21\u6570\u3002<\/p>\n<hr \/>\n<h4>7.3 Bernoulli Naive Bayes<\/h4>\n<p>\u9002\u7528\u4e8e\u4e8c\u503c\u7279\u5f81&#xff0c;\u4f8b\u5982\u67d0\u4e2a\u8bcd\u662f\u5426\u51fa\u73b0\u3001\u67d0\u9879\u5c5e\u6027\u662f\u5426\u5b58\u5728\u3002<\/p>\n<p>\u5355\u4e2a\u7279\u5f81\u7684\u6761\u4ef6\u6982\u7387\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;pkjxj(1\u2212pkj)1\u2212xjP(x_j|C_k)&#061;p_{kj}^{x_j}(1-p_{kj})^{1-x_j}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2809em;vertical-align: -0.4374em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8435em\"><span class=\"\" style=\"top: -2.3987em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2421em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3281em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2819em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4374em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1502em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\"><span class=\"mclose\">)<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3281em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2819em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">xj\u2208{0,1}x_j\\\\in\\\\{0,1\\\\}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8252em;vertical-align: -0.2861em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">{<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>Bernoulli Naive Bayes \u5173\u6ce8\u7684\u662f\u201c\u662f\u5426\u51fa\u73b0\u201d&#xff0c;\u800c Multinomial Naive Bayes \u66f4\u5173\u6ce8\u201c\u51fa\u73b0\u591a\u5c11\u6b21\u201d\u3002<\/p>\n<hr \/>\n<h4>7.4 Categorical Naive Bayes<\/h4>\n<p>\u9002\u7528\u4e8e\u79bb\u6563\u7c7b\u522b\u7279\u5f81&#xff0c;\u4f8b\u5982\u989c\u8272\u3001\u804c\u4e1a\u3001\u5730\u533a\u548c\u8bbe\u5907\u7c7b\u578b\u3002<\/p>\n<p>\u5bf9\u4e8e\u7c7b\u522b\u7279\u5f81\u53d6\u503c v&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj&#061;v\u2223Ck)&#061;count(xj&#061;v,Ck)count(Ck)P(x_j&#061;v|C_k)&#061;\\\\frac{\\\\text{count}(x_j&#061;v,C_k)}{\\\\text{count}(C_k)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord\">count<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord\">count<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5982\u679c\u67d0\u4e2a\u7ec4\u5408\u5728\u8bad\u7ec3\u96c6\u4e2d\u6ca1\u6709\u51fa\u73b0&#xff0c;\u76f4\u63a5\u4f30\u8ba1\u4f1a\u5f97\u5230\u96f6\u6982\u7387&#xff0c;\u56e0\u6b64\u901a\u5e38\u9700\u8981\u5e73\u6ed1\u3002<\/p>\n<hr \/>\n<h3>\u516b\u3001\u62c9\u666e\u62c9\u65af\u5e73\u6ed1<\/h3>\n<p>\u5047\u8bbe\u67d0\u4e2a\u8bcd\u4ece\u672a\u5728\u7c7b\u522b C_k \u4e2d\u51fa\u73b0&#xff0c;\u5219&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;0P(x_j|C_k)&#061;0<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u4e3a Naive Bayes \u4f7f\u7528\u6982\u7387\u4e58\u79ef&#xff0c;\u53ea\u8981\u4e00\u4e2a\u7279\u5f81\u6982\u7387\u4e3a\u96f6&#xff0c;\u6574\u4e2a\u7c7b\u522b\u5f97\u5206\u5c31\u4f1a\u53d8\u6210\u96f6\u3002<\/p>\n<p>\u62c9\u666e\u62c9\u65af\u5e73\u6ed1\u901a\u8fc7\u52a0\u5165\u4f2a\u8ba1\u6570\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;Nkj&#043;\u03b1Nk&#043;\u03b1VP(x_j|C_k)&#061;\\\\frac{N_{kj}&#043;\\\\alpha}{N_k&#043;\\\\alpha V}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1963em;vertical-align: -0.836em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0037em\">\u03b1<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0037em\">\u03b1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.836em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>N_kj&#xff1a;\u7279\u5f81 j \u5728\u7c7b\u522b k \u4e2d\u7684\u8ba1\u6570\u3002<\/li>\n<li>N_k&#xff1a;\u7c7b\u522b k \u4e0b\u7684\u603b\u8ba1\u6570\u3002<\/li>\n<li>V&#xff1a;\u7279\u5f81\u53d6\u503c\u6216\u8bcd\u8868\u5927\u5c0f\u3002<\/li>\n<li>\u03b1&#xff1a;\u5e73\u6ed1\u7cfb\u6570&#xff0c;\u5e38\u53d6 1\u3002<\/li>\n<\/ul>\n<p>\u5f53 \u03b1&#061;1 \u65f6&#xff0c;\u79f0\u4e3a\u52a0\u4e00\u5e73\u6ed1\u3002<\/p>\n<p>\u4ece\u8d1d\u53f6\u65af\u89d2\u5ea6\u770b&#xff0c;\u5e73\u6ed1\u53ef\u4ee5\u89e3\u91ca\u4e3a\u7ed9\u79bb\u6563\u5206\u5e03\u53c2\u6570\u52a0\u5165\u5148\u9a8c\u3002<\/p>\n<hr \/>\n<h3>\u4e5d\u3001Naive Bayes \u4e2d ML \u4e0e MAP \u7684\u4f4d\u7f6e<\/h3>\n<p>Naive Bayes \u6a21\u578b\u5305\u542b\u4e24\u7c7b\u53c2\u6570&#xff1a;<\/p>\n<li>\u7c7b\u522b\u5148\u9a8c\u6982\u7387\u3002<\/li>\n<li>\u7279\u5f81\u5728\u5404\u7c7b\u522b\u4e0b\u7684\u6761\u4ef6\u6982\u7387\u3002<\/li>\n<h4>9.1 \u4f7f\u7528 ML \u4f30\u8ba1\u7c7b\u522b\u5148\u9a8c<\/h4>\n<p>\u7c7b\u522b\u5148\u9a8c\u53ef\u4ee5\u7528\u9891\u7387\u4f30\u8ba1&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(Ck)&#061;NkNP(C_k)&#061;\\\\frac{N_k}{N}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0463em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>N_k \u662f\u7c7b\u522b k \u7684\u6837\u672c\u6570\u3002<\/li>\n<li>N \u662f\u603b\u6837\u672c\u6570\u3002<\/li>\n<\/ul>\n<h4>9.2 \u4f7f\u7528 ML \u4f30\u8ba1\u79bb\u6563\u6761\u4ef6\u6982\u7387<\/h4>\n<p>\u4e0d\u4f7f\u7528\u5e73\u6ed1\u65f6&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj&#061;v\u2223Ck)&#061;NkjvNkP(x_j&#061;v|C_k)&#061;\\\\frac{N_{kjv}}{N_k}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">v<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1963em;vertical-align: -0.836em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">v<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.836em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u662f\u57fa\u4e8e\u8bad\u7ec3\u6570\u636e\u9891\u7387\u7684\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u3002<\/p>\n<h4>9.3 \u4f7f\u7528 MAP \u4f30\u8ba1\u79bb\u6563\u6761\u4ef6\u6982\u7387<\/h4>\n<p>\u5f53\u7ed9\u591a\u9879\u5f0f\u5206\u5e03\u53c2\u6570\u52a0\u5165 Dirichlet \u5148\u9a8c\u65f6&#xff0c;\u53ef\u4ee5\u5f97\u5230\u5e26\u5e73\u6ed1\u7684\u53c2\u6570\u4f30\u8ba1\u3002<\/p>\n<p>\u5bf9\u79f0 Dirichlet \u5148\u9a8c\u4e0b&#xff0c;\u5e38\u89c1\u5f62\u5f0f\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(xj\u2223Ck)&#061;Nkj&#043;\u03b1Nk&#043;\u03b1VP(x_j|C_k)&#061;\\\\frac{N_{kj}&#043;\\\\alpha}{N_k&#043;\\\\alpha V}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0361em;vertical-align: -0.2861em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1963em;vertical-align: -0.836em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3603em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0037em\">\u03b1<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">V<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.109em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0037em\">\u03b1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.836em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u56e0\u6b64&#xff0c;\u62c9\u666e\u62c9\u65af\u5e73\u6ed1\u4e0d\u4ec5\u662f\u6570\u503c\u6280\u5de7&#xff0c;\u4e5f\u53ef\u4ee5\u770b\u6210\u4e00\u79cd MAP \u6216\u8d1d\u53f6\u65af\u53c2\u6570\u4f30\u8ba1\u601d\u60f3\u3002<\/p>\n<hr \/>\n<h3>\u5341\u3001\u624b\u5199 Gaussian Naive Bayes<\/h3>\n<p>\u4e0b\u9762\u4f7f\u7528 NumPy \u5b9e\u73b0\u4e00\u4e2a\u7b80\u5355\u7684 Gaussian Naive Bayes\u3002<\/p>\n<p><span class=\"token keyword\">import<\/span> numpy <span class=\"token keyword\">as<\/span> np<\/p>\n<p><span class=\"token keyword\">class<\/span> <span class=\"token class-name\">GaussianNaiveBayes<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token keyword\">def<\/span> <span class=\"token function\">fit<\/span><span class=\"token punctuation\">(<\/span>self<span class=\"token punctuation\">,<\/span> X<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">,<\/span> y<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token string\">&#034;GaussianNaiveBayes&#034;<\/span><span class=\"token punctuation\">:<\/span><br \/>\n        self<span class=\"token punctuation\">.<\/span>classes_ <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>unique<span class=\"token punctuation\">(<\/span>y<span class=\"token punctuation\">)<\/span><br \/>\n        n_features <span class=\"token operator\">&#061;<\/span> X<span class=\"token punctuation\">.<\/span>shape<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><\/p>\n<p>        self<span class=\"token punctuation\">.<\/span>class_prior_ <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token punctuation\">}<\/span><br \/>\n        self<span class=\"token punctuation\">.<\/span>mean_ <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token punctuation\">}<\/span><br \/>\n        self<span class=\"token punctuation\">.<\/span>var_ <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span><span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> cls <span class=\"token keyword\">in<\/span> self<span class=\"token punctuation\">.<\/span>classes_<span class=\"token punctuation\">:<\/span><br \/>\n            X_cls <span class=\"token operator\">&#061;<\/span> X<span class=\"token punctuation\">[<\/span>y <span class=\"token operator\">&#061;&#061;<\/span> cls<span class=\"token punctuation\">]<\/span><\/p>\n<p>            self<span class=\"token punctuation\">.<\/span>class_prior_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>X_cls<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>X<span class=\"token punctuation\">)<\/span><br \/>\n            self<span class=\"token punctuation\">.<\/span>mean_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> X_cls<span class=\"token punctuation\">.<\/span>mean<span class=\"token punctuation\">(<\/span>axis<span class=\"token operator\">&#061;<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            self<span class=\"token punctuation\">.<\/span>var_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> X_cls<span class=\"token punctuation\">.<\/span>var<span class=\"token punctuation\">(<\/span>axis<span class=\"token operator\">&#061;<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1e-9<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> self<\/p>\n<p>    <span class=\"token keyword\">def<\/span> <span class=\"token function\">_log_gaussian<\/span><span class=\"token punctuation\">(<\/span>self<span class=\"token punctuation\">,<\/span> X<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">,<\/span> mean<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">,<\/span> var<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token operator\">&gt;<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">:<\/span><br \/>\n        log_coefficient <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.5<\/span> <span class=\"token operator\">*<\/span> np<span class=\"token punctuation\">.<\/span>log<span class=\"token punctuation\">(<\/span><span class=\"token number\">2.0<\/span> <span class=\"token operator\">*<\/span> np<span class=\"token punctuation\">.<\/span>pi <span class=\"token operator\">*<\/span> var<span class=\"token punctuation\">)<\/span><br \/>\n        log_exponent <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">(<\/span>X <span class=\"token operator\">&#8211;<\/span> mean<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">**<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">2.0<\/span> <span class=\"token operator\">*<\/span> var<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> log_coefficient <span class=\"token operator\">&#043;<\/span> log_exponent<\/p>\n<p>    <span class=\"token keyword\">def<\/span> <span class=\"token function\">predict_log_proba<\/span><span class=\"token punctuation\">(<\/span>self<span class=\"token punctuation\">,<\/span> X<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token operator\">&gt;<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">:<\/span><br \/>\n        all_scores <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> cls <span class=\"token keyword\">in<\/span> self<span class=\"token punctuation\">.<\/span>classes_<span class=\"token punctuation\">:<\/span><br \/>\n            log_prior <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>log<span class=\"token punctuation\">(<\/span>self<span class=\"token punctuation\">.<\/span>class_prior_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            log_likelihood <span class=\"token operator\">&#061;<\/span> self<span class=\"token punctuation\">.<\/span>_log_gaussian<span class=\"token punctuation\">(<\/span><br \/>\n                X<span class=\"token punctuation\">,<\/span><br \/>\n                self<span class=\"token punctuation\">.<\/span>mean_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span><br \/>\n                self<span class=\"token punctuation\">.<\/span>var_<span class=\"token punctuation\">[<\/span>cls<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span><br \/>\n            <span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">.<\/span><span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>axis<span class=\"token operator\">&#061;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>            all_scores<span class=\"token punctuation\">.<\/span>append<span class=\"token punctuation\">(<\/span>log_prior <span class=\"token operator\">&#043;<\/span> log_likelihood<span class=\"token punctuation\">)<\/span><\/p>\n<p>        <span class=\"token keyword\">return<\/span> np<span class=\"token punctuation\">.<\/span>column_stack<span class=\"token punctuation\">(<\/span>all_scores<span class=\"token punctuation\">)<\/span><\/p>\n<p>    <span class=\"token keyword\">def<\/span> <span class=\"token function\">predict<\/span><span class=\"token punctuation\">(<\/span>self<span class=\"token punctuation\">,<\/span> X<span class=\"token punctuation\">:<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token operator\">&gt;<\/span> np<span class=\"token punctuation\">.<\/span>ndarray<span class=\"token punctuation\">:<\/span><br \/>\n        log_scores <span class=\"token operator\">&#061;<\/span> self<span class=\"token punctuation\">.<\/span>predict_log_proba<span class=\"token punctuation\">(<\/span>X<span class=\"token punctuation\">)<\/span><br \/>\n        best_indices <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>argmax<span class=\"token punctuation\">(<\/span>log_scores<span class=\"token punctuation\">,<\/span> axis<span class=\"token operator\">&#061;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> self<span class=\"token punctuation\">.<\/span>classes_<span class=\"token punctuation\">[<\/span>best_indices<span class=\"token punctuation\">]<\/span><\/p>\n<p>\u6d4b\u8bd5\u4ee3\u7801&#xff1a;<\/p>\n<p><span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>datasets <span class=\"token keyword\">import<\/span> load_iris<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>model_selection <span class=\"token keyword\">import<\/span> train_test_split<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>metrics <span class=\"token keyword\">import<\/span> accuracy_score<\/p>\n<p>X<span class=\"token punctuation\">,<\/span> y <span class=\"token operator\">&#061;<\/span> load_iris<span class=\"token punctuation\">(<\/span>return_X_y<span class=\"token operator\">&#061;<\/span><span class=\"token boolean\">True<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>X_train<span class=\"token punctuation\">,<\/span> X_test<span class=\"token punctuation\">,<\/span> y_train<span class=\"token punctuation\">,<\/span> y_test <span class=\"token operator\">&#061;<\/span> train_test_split<span class=\"token punctuation\">(<\/span><br \/>\n    X<span class=\"token punctuation\">,<\/span><br \/>\n    y<span class=\"token punctuation\">,<\/span><br \/>\n    test_size<span class=\"token operator\">&#061;<\/span><span class=\"token number\">0.25<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    random_state<span class=\"token operator\">&#061;<\/span><span class=\"token number\">42<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    stratify<span class=\"token operator\">&#061;<\/span>y<span class=\"token punctuation\">,<\/span><br \/>\n<span class=\"token punctuation\">)<\/span><\/p>\n<p>model <span class=\"token operator\">&#061;<\/span> GaussianNaiveBayes<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\nmodel<span class=\"token punctuation\">.<\/span>fit<span class=\"token punctuation\">(<\/span>X_train<span class=\"token punctuation\">,<\/span> y_train<span class=\"token punctuation\">)<\/span><\/p>\n<p>y_pred <span class=\"token operator\">&#061;<\/span> model<span class=\"token punctuation\">.<\/span>predict<span class=\"token punctuation\">(<\/span>X_test<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;Accuracy:&#034;<\/span><span class=\"token punctuation\">,<\/span> accuracy_score<span class=\"token punctuation\">(<\/span>y_test<span class=\"token punctuation\">,<\/span> y_pred<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<hr \/>\n<h3>\u5341\u4e00\u3001\u4f7f\u7528 scikit-learn<\/h3>\n<h4>11.1 Gaussian Naive Bayes<\/h4>\n<p><span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>datasets <span class=\"token keyword\">import<\/span> load_iris<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>metrics <span class=\"token keyword\">import<\/span> classification_report<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>model_selection <span class=\"token keyword\">import<\/span> train_test_split<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>naive_bayes <span class=\"token keyword\">import<\/span> GaussianNB<\/p>\n<p>X<span class=\"token punctuation\">,<\/span> y <span class=\"token operator\">&#061;<\/span> load_iris<span class=\"token punctuation\">(<\/span>return_X_y<span class=\"token operator\">&#061;<\/span><span class=\"token boolean\">True<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>X_train<span class=\"token punctuation\">,<\/span> X_test<span class=\"token punctuation\">,<\/span> y_train<span class=\"token punctuation\">,<\/span> y_test <span class=\"token operator\">&#061;<\/span> train_test_split<span class=\"token punctuation\">(<\/span><br \/>\n    X<span class=\"token punctuation\">,<\/span><br \/>\n    y<span class=\"token punctuation\">,<\/span><br \/>\n    test_size<span class=\"token operator\">&#061;<\/span><span class=\"token number\">0.25<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    random_state<span class=\"token operator\">&#061;<\/span><span class=\"token number\">42<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    stratify<span class=\"token operator\">&#061;<\/span>y<span class=\"token punctuation\">,<\/span><br \/>\n<span class=\"token punctuation\">)<\/span><\/p>\n<p>model <span class=\"token operator\">&#061;<\/span> GaussianNB<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\nmodel<span class=\"token punctuation\">.<\/span>fit<span class=\"token punctuation\">(<\/span>X_train<span class=\"token punctuation\">,<\/span> y_train<span class=\"token punctuation\">)<\/span><\/p>\n<p>y_pred <span class=\"token operator\">&#061;<\/span> model<span class=\"token punctuation\">.<\/span>predict<span class=\"token punctuation\">(<\/span>X_test<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>classification_report<span class=\"token punctuation\">(<\/span>y_test<span class=\"token punctuation\">,<\/span> y_pred<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<h4>11.2 Multinomial Naive Bayes \u6587\u672c\u5206\u7c7b<\/h4>\n<p><span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>feature_extraction<span class=\"token punctuation\">.<\/span>text <span class=\"token keyword\">import<\/span> CountVectorizer<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>naive_bayes <span class=\"token keyword\">import<\/span> MultinomialNB<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>pipeline <span class=\"token keyword\">import<\/span> Pipeline<\/p>\n<p>texts <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token string\">&#034;free prize click now&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;meeting at three pm&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;limited offer claim reward&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;project report is ready&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><\/p>\n<p>labels <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><\/p>\n<p>model <span class=\"token operator\">&#061;<\/span> Pipeline<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;vectorizer&#034;<\/span><span class=\"token punctuation\">,<\/span> CountVectorizer<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;classifier&#034;<\/span><span class=\"token punctuation\">,<\/span> MultinomialNB<span class=\"token punctuation\">(<\/span>alpha<span class=\"token operator\">&#061;<\/span><span class=\"token number\">1.0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>model<span class=\"token punctuation\">.<\/span>fit<span class=\"token punctuation\">(<\/span>texts<span class=\"token punctuation\">,<\/span> labels<span class=\"token punctuation\">)<\/span><\/p>\n<p>samples <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token string\">&#034;free reward now&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;the meeting report&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>model<span class=\"token punctuation\">.<\/span>predict<span class=\"token punctuation\">(<\/span>samples<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>model<span class=\"token punctuation\">.<\/span>predict_proba<span class=\"token punctuation\">(<\/span>samples<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>\u8fd9\u91cc alpha&#061;1.0 \u8868\u793a\u4f7f\u7528\u52a0\u4e00\u5e73\u6ed1\u3002<\/p>\n<hr \/>\n<h3>\u5341\u4e8c\u3001\u7528\u4ee3\u7801\u89c2\u5bdf ML \u4e0e MAP<\/h3>\n<p>\u4e0b\u9762\u6784\u9020\u4e00\u4e2a\u4e00\u7ef4\u9ad8\u65af\u5747\u503c\u4f30\u8ba1\u4f8b\u5b50\u3002<\/p>\n<p>\u5df2\u77e5\u89c2\u6d4b\u566a\u58f0\u65b9\u5dee\u4e3a \u03c3\u00b2&#xff0c;\u53c2\u6570\u5148\u9a8c\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bc\u223cN(\u03bc0,\u03c42)\\\\mu\\\\sim\\\\mathcal{N}(\\\\mu_0,\\\\tau^2)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u223c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1474em\">N<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1132em\">\u03c4<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6837\u672c\u5747\u503c\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">x\u02c9&#061;1N\u2211i&#061;1Nxi\\\\bar{x}&#061;\\\\frac{1}{N}\\\\sum_{i&#061;1}^{N}x_i<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5678em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.5678em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">x<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">\u02c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.106em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8283em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>ML \u4f30\u8ba1\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bc^ML&#061;x\u02c9\\\\hat{\\\\mu}_{ML}&#061;\\\\bar{x}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5678em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.5678em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">x<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">\u02c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>MAP \u4f30\u8ba1\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03bc^MAP&#061;N\u03c32x\u02c9&#043;1\u03c42\u03bc0N\u03c32&#043;1\u03c42\\\\hat{\\\\mu}_{MAP}&#061;\\\\frac{\\\\frac{N}{\\\\sigma^2}\\\\bar{x}&#043;\\\\frac{1}{\\\\tau^2}\\\\mu_0}{\\\\frac{N}{\\\\sigma^2}&#043;\\\\frac{1}{\\\\tau^2}}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">\u03bc<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.7147em;vertical-align: -1.1073em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6073em\"><span class=\"\" style=\"top: -2.2377em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8723em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7463em\"><span class=\"\" style=\"top: -2.786em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1132em\">\u03c4<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7463em\"><span class=\"\" style=\"top: -2.786em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.735em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8723em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7463em\"><span class=\"\" style=\"top: -2.786em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.5678em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">x<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2222em\"><span class=\"mord\">\u02c9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1132em\">\u03c4<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7463em\"><span class=\"\" style=\"top: -2.786em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1073em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4ee3\u7801\u5982\u4e0b&#xff1a;<\/p>\n<p><span class=\"token keyword\">import<\/span> numpy <span class=\"token keyword\">as<\/span> np<\/p>\n<p>data <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">2.1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1.9<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2.3<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>sample_mean <span class=\"token operator\">&#061;<\/span> data<span class=\"token punctuation\">.<\/span>mean<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>sigma_squared <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1.0<\/span><br \/>\nprior_mean <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0.0<\/span><br \/>\nprior_variance <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1.5<\/span><br \/>\nn <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>data<span class=\"token punctuation\">)<\/span><\/p>\n<p>ml_estimate <span class=\"token operator\">&#061;<\/span> sample_mean<\/p>\n<p>map_estimate <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">(<\/span><br \/>\n    <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">\/<\/span> sigma_squared<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">*<\/span> sample_mean<br \/>\n    <span class=\"token operator\">&#043;<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">1.0<\/span> <span class=\"token operator\">\/<\/span> prior_variance<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">*<\/span> prior_mean<br \/>\n<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token punctuation\">(<\/span><br \/>\n    <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">\/<\/span> sigma_squared<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token operator\">&#043;<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">1.0<\/span> <span class=\"token operator\">\/<\/span> prior_variance<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;ML estimate:&#034;<\/span><span class=\"token punctuation\">,<\/span> ml_estimate<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;MAP estimate:&#034;<\/span><span class=\"token punctuation\">,<\/span> map_estimate<span class=\"token punctuation\">)<\/span><\/p>\n<p>\u7531\u4e8e\u5148\u9a8c\u5747\u503c\u4e3a 0&#xff0c;MAP \u7ed3\u679c\u4f1a\u88ab\u5148\u9a8c\u5411 0 \u62c9\u8fd1\u3002<\/p>\n<p>\u5982\u679c\u589e\u52a0\u6837\u672c\u6570\u91cf&#xff0c;\u6570\u636e\u7684\u6743\u91cd\u4f1a\u53d8\u5927&#xff0c;MAP \u7ed3\u679c\u4f1a\u9010\u6e10\u63a5\u8fd1 ML\u3002<\/p>\n<hr \/>\n<h3>\u5341\u4e09\u3001Naive Bayes \u7684\u4f18\u70b9<\/h3>\n<h4>13.1 \u8bad\u7ec3\u901f\u5ea6\u5feb<\/h4>\n<p>Naive Bayes \u901a\u5e38\u53ea\u9700\u8981\u7edf\u8ba1\u5747\u503c\u3001\u65b9\u5dee\u3001\u9891\u6570\u6216\u7c7b\u522b\u6982\u7387&#xff0c;\u4e0d\u9700\u8981\u8fed\u4ee3\u4f18\u5316\u3002<\/p>\n<h4>13.2 \u9002\u5408\u9ad8\u7ef4\u7a00\u758f\u6570\u636e<\/h4>\n<p>\u5728\u6587\u672c\u5206\u7c7b\u4e2d&#xff0c;\u8bcd\u8868\u53ef\u80fd\u5305\u542b\u51e0\u4e07\u751a\u81f3\u51e0\u5341\u4e07\u4e2a\u7279\u5f81\u3002Multinomial Naive Bayes \u4ecd\u7136\u53ef\u4ee5\u9ad8\u6548\u8bad\u7ec3\u3002<\/p>\n<h4>13.3 \u5c0f\u6837\u672c\u4e0b\u8868\u73b0\u7a33\u5b9a<\/h4>\n<p>\u7531\u4e8e\u6a21\u578b\u7ed3\u6784\u7b80\u5355\u3001\u53c2\u6570\u6570\u91cf\u8f83\u5c11&#xff0c;\u5b83\u5728\u6570\u636e\u91cf\u6709\u9650\u65f6\u5e38\u6bd4\u590d\u6742\u6a21\u578b\u66f4\u7a33\u5b9a\u3002<\/p>\n<h4>13.4 \u8f93\u51fa\u6982\u7387\u5206\u6570<\/h4>\n<p>\u6a21\u578b\u53ef\u4ee5\u8f93\u51fa\u7c7b\u522b\u540e\u9a8c\u6982\u7387&#xff0c;\u4fbf\u4e8e\u6392\u5e8f\u3001\u9608\u503c\u8c03\u6574\u548c\u98ce\u9669\u8bc4\u4f30\u3002<\/p>\n<h4>13.5 \u53ef\u89e3\u91ca\u6027\u8f83\u5f3a<\/h4>\n<p>\u53ef\u4ee5\u68c0\u67e5\u7c7b\u522b\u5148\u9a8c\u3001\u8bcd\u6761\u4ef6\u6982\u7387\u3001\u5747\u503c\u548c\u65b9\u5dee&#xff0c;\u7406\u89e3\u6a21\u578b\u4e3a\u4ec0\u4e48\u505a\u51fa\u67d0\u4e2a\u5224\u65ad\u3002<\/p>\n<hr \/>\n<h3>\u5341\u56db\u3001Naive Bayes \u7684\u5c40\u9650<\/h3>\n<h4>14.1 \u6761\u4ef6\u72ec\u7acb\u5047\u8bbe\u901a\u5e38\u4e0d\u6210\u7acb<\/h4>\n<p>\u73b0\u5b9e\u4e2d\u7684\u7279\u5f81\u5f80\u5f80\u76f8\u5173\u3002\u4f8b\u5982\u6587\u672c\u4e2d\u7684\u8bcd\u5e8f\u3001\u4e0a\u4e0b\u6587\u548c\u77ed\u8bed\u5173\u7cfb\u65e0\u6cd5\u88ab\u7b80\u5355\u72ec\u7acb\u5047\u8bbe\u5b8c\u6574\u63cf\u8ff0\u3002<\/p>\n<h4>14.2 \u6982\u7387\u6821\u51c6\u53ef\u80fd\u4e0d\u7406\u60f3<\/h4>\n<p>Naive Bayes \u7684\u7c7b\u522b\u5224\u65ad\u53ef\u80fd\u51c6\u786e&#xff0c;\u4f46\u8f93\u51fa\u6982\u7387\u6709\u65f6\u8fc7\u4e8e\u81ea\u4fe1\u3002<\/p>\n<h4>14.3 \u8fde\u7eed\u7279\u5f81\u5206\u5e03\u5047\u8bbe\u53ef\u80fd\u4e0d\u5339\u914d<\/h4>\n<p>Gaussian Naive Bayes \u5047\u8bbe\u7c7b\u6761\u4ef6\u7279\u5f81\u8fd1\u4f3c\u9ad8\u65af\u5206\u5e03\u3002\u5982\u679c\u771f\u5b9e\u5206\u5e03\u660e\u663e\u504f\u659c\u3001\u591a\u5cf0\u6216\u91cd\u5c3e&#xff0c;\u6a21\u578b\u6027\u80fd\u53ef\u80fd\u4e0b\u964d\u3002<\/p>\n<h4>14.4 \u65e0\u6cd5\u81ea\u52a8\u5b66\u4e60\u590d\u6742\u7279\u5f81\u4ea4\u4e92<\/h4>\n<p>Naive Bayes \u4e0d\u4f1a\u50cf\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u90a3\u6837\u81ea\u52a8\u5b66\u4e60\u975e\u7ebf\u6027\u8868\u793a\u548c\u9ad8\u9636\u7279\u5f81\u7ec4\u5408\u3002<\/p>\n<hr \/>\n<h3>\u5341\u4e94\u3001Naive Bayes \u4e0e\u795e\u7ecf\u7f51\u7edc\u7684\u5173\u7cfb<\/h3>\n<p>Naive Bayes \u672c\u8eab\u4e0d\u662f\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b&#xff0c;\u4f46\u5b83\u4e0e\u6df1\u5ea6\u5b66\u4e60\u5b58\u5728\u591a\u65b9\u9762\u8054\u7cfb\u3002<\/p>\n<h4>15.1 \u90fd\u53ef\u4ee5\u4f7f\u7528\u6982\u7387\u5efa\u6a21<\/h4>\n<p>Naive Bayes \u663e\u5f0f\u5efa\u6a21&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(x\u2223y)P(y)P(x|y)P(y)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u795e\u7ecf\u7f51\u7edc\u5206\u7c7b\u5668\u901a\u5e38\u76f4\u63a5\u5efa\u6a21&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(y\u2223x;\u03b8)P(y|x;\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u524d\u8005\u5c5e\u4e8e\u751f\u6210\u5f0f\u5206\u7c7b\u601d\u60f3&#xff0c;\u540e\u8005\u901a\u5e38\u5c5e\u4e8e\u5224\u522b\u5f0f\u5206\u7c7b\u601d\u60f3\u3002<\/p>\n<h4>15.2 \u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3\u901a\u5e38\u91c7\u7528 ML<\/h4>\n<p>Softmax \u5206\u7c7b\u7f51\u7edc\u914d\u5408\u4ea4\u53c9\u71b5&#xff0c;\u672c\u8d28\u4e0a\u662f\u5728\u6700\u5927\u5316\u6761\u4ef6\u4f3c\u7136\u3002<\/p>\n<h4>15.3 \u6743\u91cd\u8870\u51cf\u53ef\u89e3\u91ca\u4e3a MAP<\/h4>\n<p>\u795e\u7ecf\u7f51\u7edc\u52a0\u5165 L2 \u6b63\u5219\u5316&#xff0c;\u53ef\u4ee5\u770b\u6210\u5bf9\u6743\u91cd\u65bd\u52a0\u9ad8\u65af\u5148\u9a8c\u540e\u6c42 MAP\u3002<\/p>\n<h4>15.4 \u8d1d\u53f6\u65af\u6df1\u5ea6\u5b66\u4e60\u8fdb\u4e00\u6b65\u4fdd\u7559\u53c2\u6570\u4e0d\u786e\u5b9a\u6027<\/h4>\n<p>ML \u548c MAP \u6700\u7ec8\u53ea\u7ed9\u51fa\u4e00\u4e2a\u70b9\u4f30\u8ba1&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^ML\u6216\u03b8^MAP\\\\hat{\\\\theta}_{ML}\\\\quad\\\\text{\u6216}\\\\quad\\\\hat{\\\\theta}_{MAP}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u6216<\/span><\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5b8c\u6574\u8d1d\u53f6\u65af\u63a8\u65ad\u5219\u5e0c\u671b\u5f97\u5230\u6574\u4e2a\u53c2\u6570\u540e\u9a8c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(\u03b8\u2223D)P(\\\\theta|D)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9884\u6d4b\u65f6\u5bf9\u53c2\u6570\u540e\u9a8c\u79ef\u5206&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(y\u2223x,D)&#061;\u222bP(y\u2223x,\u03b8)P(\u03b8\u2223D)d\u03b8P(y|x,D)&#061;\\\\int P(y|x,\\\\theta)P(\\\\theta|D)d\\\\theta<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.2222em;vertical-align: -0.8622em\"><\/span><span class=\"mop op-symbol large-op\" style=\"margin-right: 0.4445em;position: relative;top: -0.0011em\">\u222b<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8d1d\u53f6\u65af\u795e\u7ecf\u7f51\u7edc\u3001\u53d8\u5206\u63a8\u65ad\u548c Monte Carlo Dropout \u90fd\u4e0e\u8fd9\u79cd\u601d\u60f3\u6709\u5173\u3002<\/p>\n<hr \/>\n<h3>\u5341\u516d\u3001\u751f\u6210\u5f0f\u6a21\u578b\u4e0e\u5224\u522b\u5f0f\u6a21\u578b<\/h3>\n<p>Naive Bayes \u662f\u751f\u6210\u5f0f\u5206\u7c7b\u5668&#xff0c;\u56e0\u4e3a\u5b83\u5efa\u6a21\u8054\u5408\u5206\u5e03&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(x,y)&#061;P(y)P(x\u2223y)P(x,y)&#061;P(y)P(x|y)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9884\u6d4b\u65f6\u518d\u901a\u8fc7\u8d1d\u53f6\u65af\u5b9a\u7406\u5f97\u5230&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(y\u2223x)&#061;P(x\u2223y)P(y)P(x)P(y|x)&#061;\\\\frac{P(x|y)P(y)}{P(x)}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.363em;vertical-align: -0.936em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u903b\u8f91\u56de\u5f52\u548c\u666e\u901a Softmax \u795e\u7ecf\u7f51\u7edc\u901a\u5e38\u5c5e\u4e8e\u5224\u522b\u5f0f\u6a21\u578b&#xff0c;\u5b83\u4eec\u76f4\u63a5\u5efa\u6a21&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(y\u2223x)P(y|x)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u751f\u6210\u5f0f\u6a21\u578b\u7684\u7279\u70b9&#xff1a;<\/p>\n<ul>\n<li>\u53ef\u4ee5\u63cf\u8ff0\u6570\u636e\u5728\u4e0d\u540c\u7c7b\u522b\u4e0b\u5982\u4f55\u4ea7\u751f\u3002<\/li>\n<li>\u5728\u5c0f\u6837\u672c\u6761\u4ef6\u4e0b\u53ef\u80fd\u66f4\u9ad8\u6548\u3002<\/li>\n<li>\u53ef\u4ee5\u81ea\u7136\u5904\u7406\u7f3a\u5931\u7279\u5f81\u548c\u5148\u9a8c\u4fe1\u606f\u3002<\/li>\n<\/ul>\n<p>\u5224\u522b\u5f0f\u6a21\u578b\u7684\u7279\u70b9&#xff1a;<\/p>\n<ul>\n<li>\u76f4\u63a5\u4f18\u5316\u5206\u7c7b\u8fb9\u754c\u3002<\/li>\n<li>\u7279\u5f81\u548c\u6807\u7b7e\u5173\u7cfb\u5efa\u6a21\u66f4\u7075\u6d3b\u3002<\/li>\n<li>\u6570\u636e\u5145\u5206\u65f6\u901a\u5e38\u5177\u6709\u66f4\u9ad8\u7684\u6027\u80fd\u4e0a\u9650\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u5341\u4e03\u3001\u5e38\u89c1\u8bef\u533a<\/h3>\n<h4>\u8bef\u533a\u4e00&#xff1a;Naive Bayes \u8981\u6c42\u6240\u6709\u7279\u5f81\u5b8c\u5168\u72ec\u7acb<\/h4>\n<p>\u66f4\u51c6\u786e\u7684\u8bf4\u6cd5\u662f&#xff1a;<\/p>\n<p>Naive Bayes \u5047\u8bbe\u7279\u5f81\u5728\u7ed9\u5b9a\u7c7b\u522b\u540e\u6761\u4ef6\u72ec\u7acb\u3002<\/p>\n<p>\u65e0\u6761\u4ef6\u60c5\u51b5\u4e0b&#xff0c;\u7279\u5f81\u4ecd\u7136\u53ef\u4ee5\u76f8\u5173\u3002<\/p>\n<h4>\u8bef\u533a\u4e8c&#xff1a;ML \u8868\u793a\u6a21\u578b\u9884\u6d4b\u6982\u7387\u6700\u5927<\/h4>\n<p>ML \u6700\u5927\u5316\u7684\u662f\u8bad\u7ec3\u6570\u636e\u5728\u53c2\u6570\u4e0b\u51fa\u73b0\u7684\u6982\u7387&#xff0c;\u800c\u4e0d\u662f\u76f4\u63a5\u6700\u5927\u5316\u6d4b\u8bd5\u96c6\u51c6\u786e\u7387\u3002<\/p>\n<h4>\u8bef\u533a\u4e09&#xff1a;MAP \u5c31\u662f\u5b8c\u6574\u8d1d\u53f6\u65af\u63a8\u65ad<\/h4>\n<p>MAP \u53ea\u4fdd\u7559\u540e\u9a8c\u5206\u5e03\u7684\u6700\u9ad8\u70b9&#xff0c;\u662f\u4e00\u79cd\u70b9\u4f30\u8ba1\u3002<\/p>\n<p>\u5b8c\u6574\u8d1d\u53f6\u65af\u63a8\u65ad\u5219\u4fdd\u7559\u6574\u4e2a\u540e\u9a8c\u5206\u5e03\u3002<\/p>\n<h4>\u8bef\u533a\u56db&#xff1a;\u6b63\u5219\u5316\u53ea\u662f\u5de5\u7a0b\u6280\u5de7<\/h4>\n<p>L1 \u548c L2 \u6b63\u5219\u5316\u90fd\u53ef\u4ee5\u4ece\u53c2\u6570\u5148\u9a8c\u548c MAP \u4f30\u8ba1\u4e2d\u5f97\u5230\u6982\u7387\u89e3\u91ca\u3002<\/p>\n<h4>\u8bef\u533a\u4e94&#xff1a;\u6982\u7387\u76f8\u4e58\u65f6\u76f4\u63a5\u4f7f\u7528\u6d6e\u70b9\u6570\u5373\u53ef<\/h4>\n<p>\u5927\u91cf\u5c0f\u6982\u7387\u8fde\u7eed\u76f8\u4e58\u5bb9\u6613\u51fa\u73b0\u6570\u503c\u4e0b\u6ea2&#xff0c;\u5e94\u4f18\u5148\u5728\u5bf9\u6570\u7a7a\u95f4\u4e2d\u8ba1\u7b97\u3002<\/p>\n<hr \/>\n<h3>\u5341\u516b\u3001\u5982\u4f55\u9009\u62e9 ML\u3001MAP \u548c Naive Bayes<\/h3>\n<h4>\u9009\u62e9 ML \u7684\u60c5\u51b5<\/h4>\n<ul>\n<li>\u6570\u636e\u91cf\u8f83\u5927\u3002<\/li>\n<li>\u6ca1\u6709\u53ef\u9760\u5148\u9a8c\u3002<\/li>\n<li>\u4e3b\u8981\u76ee\u6807\u662f\u62df\u5408\u89c2\u6d4b\u6570\u636e\u3002<\/li>\n<li>\u4f7f\u7528\u6807\u51c6\u6df1\u5ea6\u5b66\u4e60\u635f\u5931\u51fd\u6570\u8bad\u7ec3\u6a21\u578b\u3002<\/li>\n<\/ul>\n<h4>\u9009\u62e9 MAP \u7684\u60c5\u51b5<\/h4>\n<ul>\n<li>\u6570\u636e\u8f83\u5c11\u3002<\/li>\n<li>\u5df2\u77e5\u53c2\u6570\u5e94\u6ee1\u8db3\u67d0\u4e9b\u5408\u7406\u7ea6\u675f\u3002<\/li>\n<li>\u5e0c\u671b\u901a\u8fc7\u5148\u9a8c\u5b9e\u73b0\u6b63\u5219\u5316\u3002<\/li>\n<li>\u9700\u8981\u628a\u9886\u57df\u77e5\u8bc6\u52a0\u5165\u53c2\u6570\u4f30\u8ba1\u3002<\/li>\n<\/ul>\n<h4>\u9009\u62e9 Naive Bayes \u7684\u60c5\u51b5<\/h4>\n<ul>\n<li>\u6587\u672c\u5206\u7c7b\u548c\u5783\u573e\u90ae\u4ef6\u8bc6\u522b\u3002<\/li>\n<li>\u9ad8\u7ef4\u7a00\u758f\u7279\u5f81\u3002<\/li>\n<li>\u8bad\u7ec3\u6570\u636e\u6709\u9650\u3002<\/li>\n<li>\u9700\u8981\u5feb\u901f\u5efa\u7acb\u57fa\u7ebf\u6a21\u578b\u3002<\/li>\n<li>\u9700\u8981\u6a21\u578b\u7b80\u5355\u3001\u6613\u89e3\u91ca\u3001\u8bad\u7ec3\u6210\u672c\u4f4e\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u5341\u4e5d\u3001\u603b\u7ed3<\/h3>\n<p>Naive Bayes\u3001ML \u548c MAP \u53ef\u4ee5\u7edf\u4e00\u5728\u6982\u7387\u5b66\u4e60\u6846\u67b6\u4e0b\u7406\u89e3\u3002<\/p>\n<p>\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u53ea\u6839\u636e\u6570\u636e\u5bfb\u627e\u53c2\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^ML&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)\\\\hat{\\\\theta}_{ML}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u540c\u65f6\u8003\u8651\u6570\u636e\u548c\u53c2\u6570\u5148\u9a8c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8^MAP&#061;arg\u2061max\u2061\u03b8P(D\u2223\u03b8)P(\u03b8)\\\\hat{\\\\theta}_{MAP}&#061;\\\\arg\\\\max_{\\\\theta}P(D|\\\\theta)P(\\\\theta)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1079em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9579em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">M<\/span><span class=\"mord mathnormal mtight\">A<\/span><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.5021em;vertical-align: -0.7521em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7521em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>Naive Bayes \u4f7f\u7528\u8d1d\u53f6\u65af\u5b9a\u7406\u548c\u6761\u4ef6\u72ec\u7acb\u5047\u8bbe\u5b8c\u6210\u5206\u7c7b&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">y^&#061;arg\u2061max\u2061CkP(Ck)\u220fj&#061;1nP(xj\u2223Ck)\\\\hat{y}&#061;\\\\arg\\\\max_{C_k}P(C_k)\\\\prod_{j&#061;1}^{n}P(x_j|C_k)<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6944em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1944em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0652em;vertical-align: -1.4138em\"><\/span><span class=\"mop\">ar<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4306em\"><span class=\"\" style=\"top: -2.3557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3448em\"><span class=\"\" style=\"top: -2.3488em;margin-left: -0.0715em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1512em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"\"><span class=\"mop\">max<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8502em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">C<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0715em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e09\u8005\u4e4b\u95f4\u7684\u6838\u5fc3\u8054\u7cfb\u662f&#xff1a;<\/p>\n<ul>\n<li>Naive Bayes \u9700\u8981\u4f30\u8ba1\u7c7b\u522b\u5148\u9a8c\u548c\u6761\u4ef6\u6982\u7387\u3002<\/li>\n<li>\u8fd9\u4e9b\u53c2\u6570\u53ef\u4ee5\u4f7f\u7528 ML \u4f30\u8ba1\u3002<\/li>\n<li>\u52a0\u5165\u5148\u9a8c\u548c\u5e73\u6ed1\u540e&#xff0c;\u53ef\u4ee5\u5f62\u6210 MAP \u4f30\u8ba1\u3002<\/li>\n<li>\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u4ea4\u53c9\u71b5\u8bad\u7ec3\u5bf9\u5e94 ML\u3002<\/li>\n<li>L1 \u548c L2 \u6b63\u5219\u5316\u53ef\u4ee5\u4ece MAP \u7684\u5148\u9a8c\u89e3\u91ca\u4e2d\u5f97\u5230\u3002<\/li>\n<\/ul>\n<p>\u638c\u63e1\u8fd9\u4e9b\u6982\u5ff5\u540e&#xff0c;\u53ef\u4ee5\u66f4\u5bb9\u6613\u7406\u89e3\u4ea4\u53c9\u71b5\u3001\u6b63\u5219\u5316\u3001\u751f\u6210\u5f0f\u6a21\u578b\u3001\u8d1d\u53f6\u65af\u795e\u7ecf\u7f51\u7edc\u4ee5\u53ca\u6982\u7387\u673a\u5668\u5b66\u4e60\u7684\u6574\u4f53\u6846\u67b6\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Naive Bayes\u3001\u6700\u5927\u4f3c\u7136\u4f30\u8ba1\u4e0e\u6700\u5927\u540e\u9a8c\u4f30\u8ba1\u672c\u6587\u7cfb\u7edf\u4ecb\u7ecd\u4e09\u4e2a\u7d27\u5bc6\u76f8\u5173\u7684\u6982\u7387\u7edf\u8ba1\u6982\u5ff5&#xff1a;Naive Bayes\u3001Maximum Likelihood Estimation&#xff08;ML&#xff09;\u4e0e Maximum A Posteriori Estimation&#xff08;MAP&#xff09;\u3002\u5b83\u4eec\u4e0d\u4ec5\u662f\u4f20\u7edf\u673a\u5668\u5b66\u4e60\u7684\u91cd\u8981\u57fa\u7840&#xff0c;\u4e5f\u6784\u6210\u4e86\u7406\u89e3\u6982\u7387\u6a21\u578b\u3001\u8d1d\u53f6\u65af\u5b66\u4e60\u3001\u6b63\u5219\u5316\u548c\u6df1\u5ea6\u5b66\u4e60\u8bad\u7ec3\u76ee\u6807\u7684\u91cd\u8981\u5165\u53e3\u3002\u4e00\u3001\u4e3a\u4ec0\u4e48\u8981\u540c\u65f6\u5b66\u4e60 Naive <\/p>\n","protected":false},"author":2,"featured_media":90832,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[10343,207,86],"topic":[],"class_list":["post-90835","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-10343","tag-207","tag-86"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ 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