{"id":113696,"date":"2026-10-07T07:09:13","date_gmt":"2026-10-06T23:09:13","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/113696.html"},"modified":"2026-10-07T07:09:13","modified_gmt":"2026-10-06T23:09:13","slug":"23%e3%80%81%e3%80%90%e6%95%b0%e5%ad%a6%e3%80%91%e3%80%90%e4%bf%a1%e6%81%af%e8%ae%ba%e3%80%91%e4%ba%a4%e5%8f%89%e7%86%b5%e4%b8%ba%e4%bb%80%e4%b9%88%e6%98%af-plogq-%e8%80%8c%e4%b8%8d%e6%98%af-qlogp","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/113696.html","title":{"rendered":"23\u3001\u3010\u6570\u5b66\u3011\u3010\u4fe1\u606f\u8bba\u3011\u4ea4\u53c9\u71b5\u4e3a\u4ec0\u4e48\u662f plog(q) \u800c\u4e0d\u662f qlog(p)\uff1a\u65b9\u5411\u3001MLE \u4e0e\u524d\u5411 KL"},"content":{"rendered":"<p>\u3010\u58f0\u660e\u3011\u672c\u535a\u5ba2\u6240\u6709\u5185\u5bb9\u5747\u4e3a\u4e2a\u4eba\u4e1a\u4f59\u65f6\u95f4\u521b\u4f5c&#xff0c;\u6240\u8ff0\u6280\u672f\u6848\u4f8b\u5747\u6765\u81ea\u516c\u5f00\u5f00\u6e90\u9879\u76ee&#xff08;\u5982Github&#xff0c;Apache\u57fa\u91d1\u4f1a&#xff09;&#xff0c;\u4e0d\u6d89\u53ca\u4efb\u4f55\u4f01\u4e1a\u673a\u5bc6\u6216\u672a\u516c\u5f00\u6280\u672f&#xff0c;\u5982\u6709\u4fb5\u6743\u8bf7\u8054\u7cfb\u5220\u9664<\/p>\n<h2>\u6807\u9898<\/h2>\n<p>23\u3001\u3010\u6570\u5b66\u3011\u3010\u4fe1\u606f\u8bba\u3011\u4ea4\u53c9\u71b5\u4e3a\u4ec0\u4e48\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        p<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        q<\/p>\n<p>       p\\\\log q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">p<\/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: 0.0359em\">q<\/span><\/span><\/span><\/span><\/span> \u800c\u4e0d\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        q<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        p<\/p>\n<p>       q\\\\log p<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/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\">p<\/span><\/span><\/span><\/span><\/span>&#xff1a;\u65b9\u5411\u3001MLE \u4e0e\u524d\u5411 KL<\/p>\n<h2>\u80cc\u666f<\/h2>\n<p>\u4e0a\u7bc7 blog \u3010\u6570\u5b66\u3011\u3010\u4fe1\u606f\u8bba\u3011\u71b5\u4e3a\u4ec0\u4e48\u53ea\u80fd\u662f\u5b83&#xff1a;Shannon \u516c\u7406\u4e0e\u552f\u4e00\u6027\u5b9a\u7406 \u7528\u56db\u6761\u516c\u7406\u628a\u71b5\u7684\u5f62\u5f0f\u9489\u6b7b\u4e86&#xff1a;\u8fde\u7eed\u3001\u5bf9\u79f0\u3001\u5747\u5300\u5206\u5e03\u968f\u7c7b\u6570\u5355\u8c03\u589e\u3001\u5206\u7ec4\u9012\u63a8\u3002\u63a8\u5bfc\u5206\u4e09\u6b65\u2014\u2014\u5206\u7ec4\u9012\u63a8\u5148\u7ed9\u51fa\u5747\u5300\u71b5\u7684\u53ef\u52a0\u6027 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        A<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        m<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        A<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>        A<\/p>\n<p>        (<\/p>\n<p>        m<\/p>\n<p>        )<\/p>\n<p>       A(nm)&#061;A(n)&#043;A(m)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">nm<\/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\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5355\u8c03\u6027\u5939\u903c\u89e3\u51fa <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        A<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        K<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        n<\/p>\n<p>       A(n)&#061;K\\\\log n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/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: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/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\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u518d\u7528\u4e00\u6b21\u5206\u7ec4\u9012\u63a8\u628a\u5747\u5300\u63a8\u5e7f\u5230\u4efb\u610f\u5206\u5e03&#xff0c;\u5f97\u5230\u552f\u4e00\u5f62\u5f0f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        &#061;<\/p>\n<p>        \u2212<\/p>\n<p>        K<\/p>\n<p>        \u2211<\/p>\n<p>         p<\/p>\n<p>         i<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>         p<\/p>\n<p>         i<\/p>\n<p>       H&#061;-K\\\\sum p_i\\\\log p_i<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/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\">\u2212<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/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: 0.3117em\"><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: 0.1667em\"><\/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\">p<\/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>&#xff1b;\u6700\u540e\u9760\u8fde\u7eed\u6027\u4ece\u6709\u7406\u6982\u7387\u94fa\u5230\u5168\u4f53\u5b9e\u6570\u3002\u7ed3\u8bba\u662f&#xff1a;\u71b5\u957f\u6210 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        \u2211<\/p>\n<p>        p<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        p<\/p>\n<p>       -\\\\sum p\\\\log p<\/p>\n<p>    <\/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 op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">p<\/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\">p<\/span><\/span><\/span><\/span><\/span> \u4e0d\u662f\u504f\u597d&#xff0c;\u800c\u662f\u88ab\u516c\u7406\u552f\u4e00\u786e\u5b9a\u7684\u3002<\/p>\n<p>\u5230\u6b64\u4e3a\u6b62&#xff0c;\u8ba8\u8bba\u7684\u90fd\u662f\u4e00\u4e2a\u5206\u5e03\u81ea\u5df1\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u4f46\u73b0\u5b9e\u91cc\u5f80\u5f80\u6709\u4e24\u4e2a\u5206\u5e03&#xff1a;\u4e00\u4e2a\u662f\u6570\u636e\u80cc\u540e\u7684\u771f\u5b9e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>        P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4e00\u4e2a\u662f\u6a21\u578b\u7ed9\u51fa\u7684\u9884\u6d4b\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>        Q<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>\u3002\u672c\u7bc7\u8981\u5ea6\u91cf\u7684\u6b63\u662f&#034;\u7528 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u53bb\u63cf\u8ff0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u6709\u591a\u8d39\u52b2&#034;\u2014\u2014\u8fd9\u5c31\u662f\u4ea4\u53c9\u71b5&#xff0c;\u800c\u5b83\u7684\u65b9\u5411\u95ee\u9898&#xff08;\u4e3a\u4ec0\u4e48\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>       P\\\\log Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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\">Q<\/span><\/span><\/span><\/span><\/span>\u3001\u4e0d\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        P<\/p>\n<p>       Q\\\\log P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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><\/span><\/span><\/span>&#xff09;\u662f\u672c\u7bc7\u7684\u6838\u5fc3\u3002<\/p>\n<hr \/>\n<h2>\u4ece&#034;\u4e00\u4e2a\u5206\u5e03&#034;\u5230&#034;\u4e24\u4e2a\u5206\u5e03&#034;<\/h2>\n<p>\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        \u2212<\/p>\n<p>         \u2211<\/p>\n<p>         x<\/p>\n<p>        P<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        P<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>       H(P)&#061;-\\\\sum_x P(x)\\\\log P(x)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/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\">x<\/span><span class=\"mclose\">)<\/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\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684\u8f93\u5165\u53ea\u6709\u4e00\u4e2a\u5206\u5e03&#xff1a;\u5b83\u56de\u7b54&#034;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u81ea\u5df1\u5e73\u5747\u6709\u591a\u4e0d\u786e\u5b9a&#034;\u3002<\/p>\n<p>\u53ef\u662f\u5728\u9884\u6d4b\u4efb\u52a1\u91cc&#xff0c;\u771f\u76f8\u548c\u731c\u6d4b\u662f\u5206\u5f00\u7684&#xff1a;\u771f\u5b9e\u6807\u7b7e\u670d\u4ece\u67d0\u4e2a\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6a21\u578b\u5410\u51fa\u4e00\u4e2a\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>\u3002\u60f3\u77e5\u9053&#034;\u6a21\u578b\u79bb\u771f\u76f8\u6709\u591a\u8fdc&#034;&#xff0c;\u5c31\u9700\u8981\u4e00\u4e2a\u540c\u65f6\u5403\u8fdb\u4e24\u4e2a\u5206\u5e03\u7684\u91cf\u3002\u6700\u76f4\u63a5\u7684\u60f3\u6cd5\u662f&#xff1a;\u62ff <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>        Q<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u53bb\u6c47\u62a5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>        P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u4e16\u754c\u91cc\u771f\u5b9e\u53d1\u751f\u7684\u4e8b&#xff0c;\u5e73\u5747\u4f1a\u6709\u591a\u610f\u5916&#xff1f; \u8fd9\u4e2a&#034;\u5e73\u5747\u610f\u5916&#034;\u5c31\u662f\u4ea4\u53c9\u71b5\u3002<\/p>\n<hr \/>\n<h2>\u4ea4\u53c9\u71b5\u7684\u5b9a\u4e49&#xff1a;\u8c01\u5f53\u6743\u91cd&#xff0c;\u8c01\u8fdb log<\/h2>\n<p>\u4ea4\u53c9\u71b5&#xff08;cross-entropy&#xff09; \u5b9a\u4e49\u4e3a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2212<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          E<\/p>\n<p>           x<\/p>\n<p>           \u223c<\/p>\n<p>           P<\/p>\n<p>         [<\/p>\n<p>         \u2212<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         ]<\/p>\n<p>        H(P,Q)&#061;-\\\\sum_x P(x)\\\\log Q(x)&#061;\\\\mathbb E_{x\\\\sim P}\\\\big[-\\\\log Q(x)\\\\big]<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 2.3em;vertical-align: -1.25em\"><\/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: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/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: 1.2em;vertical-align: -0.35em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">E<\/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\">x<\/span><span class=\"mrel mtight\">\u223c<\/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=\"mord\"><span class=\"delimsizing size1\">[<\/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: 1.2em;vertical-align: -0.35em\"><\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"delimsizing size1\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9010\u9879\u8bfb\u8fd9\u4e2a\u5f0f\u5b50&#xff0c;\u5b83\u4f9d\u7136\u662f&#034;\u4fe1\u606f\u91cf\u7684\u52a0\u6743\u5e73\u5747&#034;&#xff1a;<\/p>\n<ul>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         \u2212<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>        -\\\\log Q(x)<\/p>\n<p>     <\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1a;\u7ed3\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>        x<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span> \u7684\u4fe1\u606f\u91cf&#xff0c;\u7528\u7684\u662f\u9884\u6d4b\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>        Q<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>        P(x)<\/p>\n<p>     <\/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=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1a;\u6743\u91cd&#xff0c;\u662f\u7ed3\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>        x<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span> \u5728\u73b0\u5b9e\u91cc\u7684\u771f\u5b9e\u9891\u7387&#xff1b;<\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>        \\\\sum_x<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1a;\u5bf9\u6240\u6709\u7ed3\u679c\u6c42\u5e73\u5747\u3002<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261006230911-6ac57f97326f2.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u56fe 1 \u5de6\u8fb9\u662f\u4e24\u4e2a\u5206\u5e03\u7684\u5bf9\u6bd4&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u662f\u771f\u76f8\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u662f\u9884\u6d4b&#xff09;&#xff0c;\u53f3\u8fb9\u628a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>       -\\\\log Q(x)<\/p>\n<p>    <\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u753b\u6210\u67f1\u3001\u5e76\u5728\u6bcf\u6839\u67f1\u4e0a\u6807\u51fa\u5b83\u5bf9\u5e94\u7684\u6743\u91cd <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>       P(x)<\/p>\n<p>    <\/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=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u5206\u5de5\u975e\u5e38\u660e\u786e&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>        P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u5f53\u6743\u91cd&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>        Q<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u8fdb <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>        \\\\log<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u5e26\u4e00\u7ec4\u6570\u5b57\u4f53\u4f1a\u3002\u8bbe <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        &#061;<\/p>\n<p>        [<\/p>\n<p>        0.7<\/p>\n<p>        ,<\/p>\n<p>        0.2<\/p>\n<p>        ,<\/p>\n<p>        0.1<\/p>\n<p>        ]<\/p>\n<p>       P&#061;[0.7,0.2,0.1]<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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\">0.7<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.2<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.1<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span>\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>        &#061;<\/p>\n<p>        [<\/p>\n<p>        0.5<\/p>\n<p>        ,<\/p>\n<p>        0.3<\/p>\n<p>        ,<\/p>\n<p>        0.2<\/p>\n<p>        ]<\/p>\n<p>       Q&#061;[0.5,0.3,0.2]<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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\">0.5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.3<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.2<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span>&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2212<\/p>\n<p>         (<\/p>\n<p>         0.7<\/p>\n<p>         ln<\/p>\n<p>         \u2061<\/p>\n<p>         0.5<\/p>\n<p>         &#043;<\/p>\n<p>         0.2<\/p>\n<p>         ln<\/p>\n<p>         \u2061<\/p>\n<p>         0.3<\/p>\n<p>         &#043;<\/p>\n<p>         0.1<\/p>\n<p>         ln<\/p>\n<p>         \u2061<\/p>\n<p>         0.2<\/p>\n<p>         )<\/p>\n<p>         \u2248<\/p>\n<p>         0.887<\/p>\n<p>        H(P,Q)&#061;-(0.7\\\\ln0.5&#043;0.2\\\\ln0.3&#043;0.1\\\\ln0.2)\\\\approx0.887<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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\">\u2212<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">0.7<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.5<\/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: 0.7778em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">0.2<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.3<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">0.1<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">ln<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0.2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/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\">0.887<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u8d8a\u8d34\u8fd1 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5e73\u5747\u8d8a\u4e0d\u610f\u5916&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>       H(P,Q)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u8d8a\u5c0f&#xff1b;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>        &#061;<\/p>\n<p>        P<\/p>\n<p>       Q&#061;P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;\u4ea4\u53c9\u71b5\u9000\u56de\u6210\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>       H(P)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u6ce8\u610f\u5f0f\u5b50\u91cc <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u4e0e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u7684\u5730\u4f4d\u5e76\u4e0d\u5bf9\u7b49&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u51fa\u73b0\u5728\u6743\u91cd\u4f4d\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u51fa\u73b0\u5728\u5bf9\u6570\u4f4d\u3002\u4e4b\u540e\u4f1a\u770b\u5230&#xff0c;\u6b63\u662f\u8fd9\u4e2a\u4e0d\u5bf9\u7b49\u9020\u6210\u4e24\u4e2a\u65b9\u5411\u7684\u5dee\u5f02&#xff0c;\u4e5f\u51b3\u5b9a\u4e86\u8bad\u7ec3\u65f6\u8be5\u9009\u54ea\u4e2a\u65b9\u5411\u3002<\/p>\n<hr \/>\n<h2>\u71b5\u4e0e\u4ea4\u53c9\u71b5&#xff1a;\u4e00\u5f20\u8868\u770b\u6e05<\/h2>\n<p>\u4e09\u4e2a\u76f8\u5173\u7684\u91cf\u653e\u5728\u4e00\u8d77&#xff0c;\u5dee\u522b\u53ea\u5728&#034;\u8c01\u5f53\u6743\u91cd\u3001\u8c01\u8fdb log&#034;&#xff1a;<\/p>\n<table>\n<tr>\u91cf\u5b9a \u4e49\u6743 \u91cdlog \u91cc\u542b \u4e49<\/tr>\n<tbody>\n<tr>\n<td align=\"left\">\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           H<\/p>\n<p>           (<\/p>\n<p>           P<\/p>\n<p>           )<\/p>\n<p>          H(P)<\/p>\n<p>       <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           \u2212<\/p>\n<p>            \u2211<\/p>\n<p>            x<\/p>\n<p>           P<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           P<\/p>\n<p>          -\\\\sum_x P\\\\log P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/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=\"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><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u81ea\u5df1\u7684\u4e0d\u786e\u5b9a\u6027<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u4ea4\u53c9\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           H<\/p>\n<p>           (<\/p>\n<p>           P<\/p>\n<p>           ,<\/p>\n<p>           Q<\/p>\n<p>           )<\/p>\n<p>          H(P,Q)<\/p>\n<p>       <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           \u2212<\/p>\n<p>            \u2211<\/p>\n<p>            x<\/p>\n<p>           P<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           Q<\/p>\n<p>          -\\\\sum_x P\\\\log Q<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/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=\"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\">Q<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           Q<\/p>\n<p>          Q<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\">\u7528 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           Q<\/p>\n<p>          Q<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u63cf\u8ff0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u7684\u4ee3\u4ef7<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u53cd\u5411\u4ea4\u53c9\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           H<\/p>\n<p>           (<\/p>\n<p>           Q<\/p>\n<p>           ,<\/p>\n<p>           P<\/p>\n<p>           )<\/p>\n<p>          H(Q,P)<\/p>\n<p>       <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           \u2212<\/p>\n<p>            \u2211<\/p>\n<p>            x<\/p>\n<p>           Q<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           P<\/p>\n<p>          -\\\\sum_x Q\\\\log P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           Q<\/p>\n<p>          Q<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\"><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span><\/td>\n<td align=\"left\">\u7528 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           P<\/p>\n<p>          P<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u63cf\u8ff0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           Q<\/p>\n<p>          Q<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u7684\u4ee3\u4ef7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u71b5\u662f\u4ea4\u53c9\u71b5\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>        &#061;<\/p>\n<p>        P<\/p>\n<p>       Q&#061;P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u65f6\u7684\u7279\u4f8b&#xff1b;\u4e24\u4e2a\u65b9\u5411\u7684\u4ea4\u53c9\u71b5\u53ea\u6709\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        &#061;<\/p>\n<p>        Q<\/p>\n<p>       P&#061;Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u65f6\u624d\u76f8\u7b49\u3002\u8bb0\u4f4f\u8fd9\u5f20\u8868&#xff0c;\u65b9\u5411\u95ee\u9898\u57fa\u672c\u4e0d\u4f1a\u518d\u9519\u3002<\/p>\n<hr \/>\n<h2>\u4e3a\u4ec0\u4e48\u6743\u91cd\u5fc5\u987b\u662f\u771f\u5b9e\u5206\u5e03 P<\/h2>\n<p>\u5173\u952e\u7684\u4e00\u95ee&#xff1a;\u65e2\u7136\u5f0f\u5b50\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        \u2211<\/p>\n<p>        P<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>       -\\\\sum P\\\\log Q<\/p>\n<p>    <\/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 op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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\">Q<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4e3a\u4ec0\u4e48\u6743\u91cd\u975e\u5f97\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>\u3001\u800c\u4e0d\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>&#xff1f;<\/p>\n<p>\u7b54\u6848\u85cf\u5728\u671f\u671b\u7684\u5199\u6cd5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         E<\/p>\n<p>          x<\/p>\n<p>          \u223c<\/p>\n<p>          P<\/p>\n<p>        [<\/p>\n<p>        \u22c5<\/p>\n<p>        ]<\/p>\n<p>       \\\\mathbb E_{x\\\\sim P}[\\\\cdot]<\/p>\n<p>    <\/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 mathbb\">E<\/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\">x<\/span><span class=\"mrel mtight\">\u223c<\/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=\"mopen\">[<\/span><span class=\"mord\">\u22c5<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span> \u91cc&#xff1a;\u6743\u91cd\u5c31\u662f&#034;\u5bf9\u8c01\u53d6\u5e73\u5747&#034;\u3002\u6837\u672c\u662f\u4ece\u771f\u5b9e\u6570\u636e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u91cc\u91c7\u51fa\u6765\u7684&#xff0c;\u6240\u4ee5&#034;\u5e73\u5747\u6709\u591a\u610f\u5916&#034;\u5fc5\u987b\u6309 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u6765\u5e73\u5747\u3002\u82e5\u628a\u6743\u91cd\u6362\u6210 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7b97\u7684\u5c31\u4e0d\u518d\u662f&#034;\u7528 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u63cf\u8ff0\u771f\u76f8\u7684\u4ee3\u4ef7&#034;&#xff0c;\u800c\u53d8\u6210\u4e86&#034;\u7528\u771f\u76f8\u63cf\u8ff0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u7684\u4ee3\u4ef7&#034;\u2014\u2014\u4e3b\u5ba2\u98a0\u5012\u3002<\/p>\n<p>\u6362\u4e2a\u8bf4\u6cd5&#xff1a;&#034;\u5b9e\u9645\u53d1\u751f\u4e86\u4ec0\u4e48&#034;\u51b3\u5b9a\u600e\u4e48\u52a0\u6743&#xff0c;&#034;\u6a21\u578b\u731c\u4e86\u4ec0\u4e48&#034;\u51b3\u5b9a\u60ca\u8bb6\u5927\u5c0f\u3002 \u524d\u8005\u662f\u6743\u91cd <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u540e\u8005\u8fdb <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>       \\\\log<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span>\u3002\u5982\u679c\u53cd\u8fc7\u6765\u5199 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        \u2211<\/p>\n<p>        Q<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        P<\/p>\n<p>       -\\\\sum Q\\\\log P<\/p>\n<p>    <\/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 op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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><\/span><\/span><\/span>&#xff0c;weight \u53d8\u6210\u4e86\u6a21\u578b\u7684\u4fe1\u5ff5\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>       \\\\log<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><\/span><\/span><\/span><\/span> \u91cc\u653e\u7684\u662f\u771f\u76f8&#xff0c;\u90a3\u662f\u5728\u95ee&#034;\u6a21\u578b\u6709\u591a\u76f8\u4fe1\u81ea\u5df1&#034;&#xff0c;\u5bf9\u8bad\u7ec3\u6ca1\u6709\u610f\u4e49\u3002<\/p>\n<hr \/>\n<h2>\u4ece\u6700\u5927\u4f3c\u7136\u5230\u4ea4\u53c9\u71b5<\/h2>\n<p>\u4e0a\u9762\u7684&#034;\u91c7\u6837\u89c6\u89d2&#034;\u53ef\u4ee5\u505a\u6210\u4e25\u683c\u63a8\u5bfc&#xff0c;\u7ed3\u8bba\u5b8c\u5168\u4e00\u81f4\u3002\u8fd9\u6b63\u662f\u6700\u5927\u4f3c\u7136\u4f30\u8ba1&#xff08;Maximum Likelihood Estimation&#xff0c;MLE&#xff09; &#xff1a;<\/p>\n<p>\u6709\u4e00\u6279\u4ece <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u91c7\u51fa\u7684\u72ec\u7acb\u6837\u672c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>         1<\/p>\n<p>        ,<\/p>\n<p>        \u2026<\/p>\n<p>        ,<\/p>\n<p>         x<\/p>\n<p>         N<\/p>\n<p>       x_1,\\\\dots,x_N<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/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=\"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><\/span><\/span><\/span>&#xff0c;\u6a21\u578b <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u5bf9\u5b83\u4eec\u7684\u5bf9\u6570\u4f3c\u7136\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>         i<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>        (<\/p>\n<p>         x<\/p>\n<p>         i<\/p>\n<p>        )<\/p>\n<p>       \\\\sum_i\\\\log Q(x_i)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.162em\"><span class=\"\" style=\"top: -2.4003em;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: 0.2997em\"><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 mathnormal\">Q<\/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=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u6309\u5927\u6570\u5b9a\u5f8b&#xff0c;\u6837\u672c\u9891\u7387\u4f1a\u6536\u655b\u5230\u771f\u5b9e\u6982\u7387&#xff0c;\u4e8e\u662f<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          1<\/p>\n<p>          N<\/p>\n<p>          \u2211<\/p>\n<p>          i<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>          x<\/p>\n<p>          i<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>          P<\/p>\n<p>          ^<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u00a0<\/p>\n<p>          \u2192<\/p>\n<p>            N<\/p>\n<p>            \u2192<\/p>\n<p>            \u221e<\/p>\n<p>         \u00a0<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>        \\\\frac1N\\\\sum_i\\\\log Q(x_i)&#061;\\\\sum_x \\\\hat P(x)\\\\log Q(x)\\\\ \\\\xrightarrow[N\\\\to\\\\infty]{}\\\\ \\\\sum_x P(x)\\\\log Q(x)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.5991em;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: 1.05em\"><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 mathnormal mtight\">i<\/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><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\">Q<\/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=\"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: 2.3em;vertical-align: -1.25em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9468em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><span class=\"\" style=\"top: -3.2523em\"><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=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel x-arrow\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.622em\"><span class=\"\" style=\"top: -3.322em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight x-arrow-pad\"><span class=\"mord mtight\"><\/span><\/span><\/span><span class=\"svg-align\" style=\"top: -2.689em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"hide-tail\" style=\"height: 0.522em;min-width: 1.469em\"><\/p>\n<p>            <\/span><\/span><span class=\"\" style=\"top: -2.0997em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight x-arrow-pad\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.109em\">N<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mtight\">\u221e<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6003em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.3em;vertical-align: -1.25em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         ^<\/p>\n<p>       \\\\hat P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9468em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9468em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><span class=\"\" style=\"top: -3.2523em\"><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><\/span><\/span><\/span> \u662f\u7ecf\u9a8c\u5206\u5e03&#xff08;\u6837\u672c\u91cc\u5404\u7ed3\u679c\u51fa\u73b0\u7684\u9891\u7387&#xff09;\u3002\u6700\u5927\u5316\u5bf9\u6570\u4f3c\u7136\u5c31\u662f\u6700\u5c0f\u5316\u5b83\u7684\u76f8\u53cd\u6570&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           max<\/p>\n<p>           \u2061<\/p>\n<p>          Q<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u27fa<\/p>\n<p>           min<\/p>\n<p>           \u2061<\/p>\n<p>          Q<\/p>\n<p>         (<\/p>\n<p>         \u2212<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>           min<\/p>\n<p>           \u2061<\/p>\n<p>          Q<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>        \\\\max_Q\\\\sum_x P(x)\\\\log Q(x)\\\\quad\\\\Longleftrightarrow\\\\quad\\\\min_Q\\\\Big(-\\\\sum_x P(x)\\\\log Q(x)\\\\Big)&#061;\\\\min_Q H(P,Q)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.3em;vertical-align: -1.25em\"><\/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 mathnormal mtight\">Q<\/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: 0.8804em\"><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: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u27fa<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0304em;vertical-align: -0.8804em\"><\/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: -2.3557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">Q<\/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: 0.8804em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">(<\/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: 2.4em;vertical-align: -1.25em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"delimsizing size2\">)<\/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: 1.6304em;vertical-align: -0.8804em\"><\/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: -2.3557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">Q<\/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: 0.8804em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261006230911-6ac57f978fb57.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u56fe 2 \u628a\u8fd9\u6761\u94fe\u753b\u6210\u6d41\u7a0b&#xff1a;\u6570\u636e\u4ece <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u91c7\u51fa \u2192 \u8ba1\u6570\u5f97\u5230\u7ecf\u9a8c\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         ^<\/p>\n<p>       \\\\hat P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9468em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9468em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><span class=\"\" style=\"top: -3.2523em\"><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><\/span><\/span><\/span> \u2192 \u5e73\u5747\u65f6\u6743\u91cd\u53ea\u80fd\u6309 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u52a0\u6743 \u2192 \u6700\u5c0f\u5316\u4ea4\u53c9\u71b5\u7b49\u4ef7\u4e8e\u6700\u5927\u4f3c\u7136\u3002\u987a\u5e26\u4e00\u63d0&#xff0c;\u8fd9\u4e2a\u76ee\u6807\u5bf9 logits \u7684\u68af\u5ea6\u5f62\u5f0f\u5f88\u5e72\u51c0&#xff08;\u7b2c 235 \u7bc7\u63d0\u8fc7&#xff09;&#xff0c;\u662f\u5b83\u9002\u5408\u505a\u8bad\u7ec3\u635f\u5931\u7684\u53e6\u4e00\u4e2a\u539f\u56e0\u3002\u63a8\u5bfc\u91cc&#034;\u6743\u91cd\u4f4d&#034;\u662f\u88ab\u91c7\u6837\u52a8\u4f5c\u81ea\u5df1\u5360\u5b9a\u7684&#xff0c;\u4e0d\u662f\u4eba\u4e3a\u9009\u62e9\u2014\u2014\u8fd9\u5c31\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>       P\\\\log Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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\">Q<\/span><\/span><\/span><\/span><\/span> \u800c\u4e0d\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        P<\/p>\n<p>       Q\\\\log P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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><\/span><\/span><\/span> \u7684\u6700\u786c\u4f9d\u636e&#xff1a;\u8bad\u7ec3\u7684\u76ee\u6807\u662f\u8ba9\u6a21\u578b\u89e3\u91ca\u6570\u636e&#xff0c;\u800c\u4e0d\u662f\u8ba9\u6570\u636e\u89e3\u91ca\u6a21\u578b\u3002<\/p>\n<hr \/>\n<h2>\u62c6\u5f00\u6765\u770b&#xff1a;\u4ea4\u53c9\u71b5 &#061; \u71b5 &#043; KL<\/h2>\n<p>\u4ea4\u53c9\u71b5\u548c\u71b5\u53ea\u5dee\u4e00\u5757&#xff0c;\u628a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>        &#061;<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        P<\/p>\n<p>        &#043;<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>         Q<\/p>\n<p>         P<\/p>\n<p>       \\\\log Q&#061;\\\\log P&#043;\\\\log\\\\frac{Q}{P}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/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\">Q<\/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: 0.8889em;vertical-align: -0.1944em\"><\/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=\"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: 1.2694em;vertical-align: -0.345em\"><\/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=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9244em\"><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 mathnormal mtight\" style=\"margin-right: 0.1389em\">P<\/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: -3.4461em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">Q<\/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><\/span> \u4ee3\u8fdb\u53bb\u5c31\u80fd\u62c6&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2212<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u2212<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>           Q<\/p>\n<p>           (<\/p>\n<p>           x<\/p>\n<p>           )<\/p>\n<p>           P<\/p>\n<p>           (<\/p>\n<p>           x<\/p>\n<p>           )<\/p>\n<p>         &#061;<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>          K<\/p>\n<p>          L<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         \u2225<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>        H(P,Q)&#061;-\\\\sum_x P(x)\\\\log P(x)-\\\\sum_x P(x)\\\\log\\\\frac{Q(x)}{P(x)}&#061;H(P)&#043;\\\\mathrm{KL}(P\\\\|Q)<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 2.3em;vertical-align: -1.25em\"><\/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: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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\">x<\/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: 2.677em;vertical-align: -1.25em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><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\">x<\/span><span class=\"mclose\">)<\/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=\"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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/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 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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261006230911-6ac57f97dea18.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u56fe 3 \u7528\u521a\u624d\u7684\u4f8b\u5b50\u628a\u5b83\u753b\u6210\u5806\u53e0\u6761&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>        \u2248<\/p>\n<p>        0.802<\/p>\n<p>       H(P)\\\\approx0.802<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/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\">0.802<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u518d\u52a0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         K<\/p>\n<p>         L<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        \u2225<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>        \u2248<\/p>\n<p>        0.085<\/p>\n<p>       \\\\mathrm{KL}(P\\\\|Q)\\\\approx0.085<\/p>\n<p>    <\/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 mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/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\">0.085<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5f97\u5230\u4ea4\u53c9\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        0.887<\/p>\n<p>       0.887<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0.887<\/span><\/span><\/span><\/span><\/span>\u3002\u56e0\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>       H(P)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u662f\u771f\u5b9e\u5206\u5e03\u81ea\u5df1\u7684\u3001\u8bad\u7ec3\u65f6\u6539\u4e0d\u52a8&#xff0c;\u6240\u4ee5<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u6700\u5c0f\u5316\u00a0<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         \u27fa<\/p>\n<p>         \u6700\u5c0f\u5316\u00a0<\/p>\n<p>          K<\/p>\n<p>          L<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         \u2225<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>        \\\\text{\u6700\u5c0f\u5316 }H(P,Q)\\\\quad\\\\Longleftrightarrow\\\\quad\\\\text{\u6700\u5c0f\u5316 }\\\\mathrm{KL}(P\\\\|Q)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u6700\u5c0f\u5316<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u27fa<\/span><span class=\"mspace\" style=\"margin-right: 1em\"><\/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 text\"><span class=\"mord cjk_fallback\">\u6700\u5c0f\u5316<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e5f\u5c31\u662f\u8ba9\u9884\u6d4b\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u65e0\u9650\u8d34\u8fd1\u771f\u5b9e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>\u3002\u8fd9\u5c31\u662f\u8bed\u8a00\u6a21\u578b\u8bad\u7ec3\u5728\u505a\u7684\u4e8b\u3002<\/p>\n<hr \/>\n<h2>one-hot \u7279\u4f8b&#xff1a;\u584c\u7f29\u6210\u4e00\u9879<\/h2>\n<p>\u7b2c 13 \u7bc7\u8bb2\u8fc7\u7684\u584c\u7f29&#xff0c;\u5728\u8fd9\u91cc\u518d\u770b\u4e00\u6b21\u4f1a\u66f4\u6709\u611f\u89e6\u3002\u8bed\u8a00\u6a21\u578b\u7684\u771f\u5b9e\u6807\u7b7e\u662f one-hot&#xff08;\u72ec\u70ed&#xff09; \u5411\u91cf&#xff1a;\u6b63\u786e token \u90a3\u4e00\u7ef4\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        1<\/p>\n<p>       1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>\u3001\u5176\u4f59\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        0<\/p>\n<p>       0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>\u3002\u628a\u5b83\u4ee3\u8fdb\u4ea4\u53c9\u71b5\u6c42\u548c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         1<\/p>\n<p>         \u00d7<\/p>\n<p>         (<\/p>\n<p>         \u2212<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         \u6b63\u786e<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>          \u2211<\/p>\n<p>          \u5176\u5b83<\/p>\n<p>         0<\/p>\n<p>         \u00d7<\/p>\n<p>         (<\/p>\n<p>         \u22ef<br \/>\n        \u2009<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2212<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         \u6b63\u786e<\/p>\n<p>         )<\/p>\n<p>        H(P,Q)&#061;1\\\\times\\\\big(-\\\\log Q(\\\\text{\u6b63\u786e})\\\\big)&#043;\\\\sum_{\\\\text{\u5176\u5b83}}0\\\\times(\\\\cdots)&#061;-\\\\log Q(\\\\text{\u6b63\u786e})<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/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: 1.2em;vertical-align: -0.35em\"><\/span><span class=\"mord\"><span class=\"delimsizing size1\">(<\/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: 1.2em;vertical-align: -0.35em\"><\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u6b63\u786e<\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mord\"><span class=\"delimsizing size1\">)<\/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: 2.3443em;vertical-align: -1.2943em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8557em;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 text mtight\"><span class=\"mord cjk_fallback mtight\">\u5176\u5b83<\/span><\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2943em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/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\">\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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u6b63\u786e<\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u9664\u6b63\u786e\u9879\u4ee5\u5916&#xff0c;\u6240\u6709\u9879\u90fd\u88ab\u6743\u91cd <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        &#061;<\/p>\n<p>        0<\/p>\n<p>       P&#061;0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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> \u4e58\u6ca1\u4e86&#xff0c;\u6c42\u548c\u584c\u7f29\u6210\u4e00\u9879&#xff0c;\u5373\u8d1f\u5bf9\u6570\u4f3c\u7136&#xff08;Negative Log-Likelihood&#xff0c;NLL&#xff09; \u3002\u8fd9\u89e3\u91ca\u4e86\u8bad\u7ec3\u91cc\u4e00\u4e2a\u5e38\u89c1\u73b0\u8c61&#xff1a;\u4e00\u4e2a\u4f4d\u7f6e\u7684\u635f\u5931\u53ea\u53d6\u51b3\u4e8e\u6a21\u578b\u7ed9\u6b63\u786e token \u7684\u6982\u7387&#xff0c;\u7ed9\u5f97\u8d8a\u9ad8\u3001\u635f\u5931\u8d8a\u5c0f\u3002<\/p>\n<p>\u8fd9\u4e5f\u6b63\u662f\u7b2c 13\u300114 \u7bc7\u90a3\u6761\u94fe\u7684\u843d\u70b9&#xff1a;one-hot \u8ba9&#034;\u5b8c\u6574\u7684\u4ea4\u53c9\u71b5&#034;\u9000\u5316\u6210&#034;\u53ea\u770b\u6b63\u786e\u7b54\u6848\u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>       -\\\\log Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/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\">Q<\/span><\/span><\/span><\/span><\/span>&#034;&#xff0c;\u800c\u84b8\u998f\u7b49\u573a\u666f\u91cc\u6559\u5e08\u7ed9\u7684\u8f6f\u6807\u7b7e\u4e0d\u662f one-hot&#xff0c;\u6b64\u65f6\u5b8c\u6574\u5f62\u5f0f&#xff08;\u6c42\u548c\u4e0d\u584c\u7f29&#xff09;\u624d\u91cd\u65b0\u6d3e\u4e0a\u7528\u573a\u3002<\/p>\n<hr \/>\n<h2>\u53cd\u8fc7\u6765\u5199\u4f1a\u600e\u6837<\/h2>\n<p>\u5982\u679c\u504f\u8981\u628a\u65b9\u5411\u5012\u8fc7\u6765&#xff0c;\u5f97\u5230\u7684\u662f\u53cd\u5411\u4ea4\u53c9\u71b5&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         Q<\/p>\n<p>         ,<\/p>\n<p>         P<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2212<\/p>\n<p>          \u2211<\/p>\n<p>          x<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>          K<\/p>\n<p>          L<\/p>\n<p>         (<\/p>\n<p>         Q<\/p>\n<p>         \u2225<\/p>\n<p>         P<\/p>\n<p>         )<\/p>\n<p>        H(Q,P)&#061;-\\\\sum_x Q(x)\\\\log P(x)&#061;H(Q)&#043;\\\\mathrm{KL}(Q\\\\|P)<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 2.3em;vertical-align: -1.25em\"><\/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: 1.05em\"><span class=\"\" style=\"top: -1.9em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/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><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.25em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/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\">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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5b83\u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>       H(P,Q)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e00\u822c\u4e0d\u76f8\u7b49\u3002\u540c\u4e00\u4e2a\u4f8b\u5b50&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>        \u2248<\/p>\n<p>        0.887<\/p>\n<p>       H(P,Q)\\\\approx0.887<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/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\">0.887<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u800c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        Q<\/p>\n<p>        ,<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>        \u2248<\/p>\n<p>        1.122<\/p>\n<p>       H(Q,P)\\\\approx1.122<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/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\">1.122<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261006230912-6ac57f980a0b7.png\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>\u56fe 4 \u5bf9\u6bd4\u4e24\u4e2a\u65b9\u5411&#xff1a;\u524d\u5411\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>         K<\/p>\n<p>         L<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        \u2225<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>       H(P)&#043;\\\\mathrm{KL}(P\\\\|Q)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u53cd\u5411\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>         K<\/p>\n<p>         L<\/p>\n<p>        (<\/p>\n<p>        Q<\/p>\n<p>        \u2225<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>       H(Q)&#043;\\\\mathrm{KL}(Q\\\\|P)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6570\u503c\u4e0d\u540c\u3002\u53cd\u5411\u9664\u4e86&#034;\u4e3b\u5ba2\u98a0\u5012&#034;&#xff0c;\u8fd8\u6709\u4e00\u4e2a\u81f4\u547d\u95ee\u9898&#xff1a;\u5f53\u771f\u5b9e\u6807\u7b7e\u662f one-hot&#xff08;\u7b2c 13\u300114 \u7bc7&#xff09;\u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>        P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u5728\u5f88\u591a\u7ef4\u4e0a\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         0<\/p>\n<p>        0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u800c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2212<\/p>\n<p>         \u2211<\/p>\n<p>         Q<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         P<\/p>\n<p>        -\\\\sum Q\\\\log P<\/p>\n<p>     <\/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 op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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><\/span><\/span><\/span> \u91cc\u4f1a\u51fa\u73b0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         0<\/p>\n<p>        \\\\log 0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/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\">0<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u76f4\u63a5\u53d1\u6563\u2014\u2014\u8fd9\u6837\u7684\u76ee\u6807\u6839\u672c\u6ca1\u6cd5\u8bad\u7ec3\u3002\u800c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>       H(P,Q)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u91cc <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        &#061;<\/p>\n<p>        0<\/p>\n<p>       P&#061;0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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> \u7684\u9879\u88ab\u6743\u91cd\u4e58\u6ca1&#xff0c;\u6070\u597d\u584c\u7f29\u6210 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2212<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>        (<\/p>\n<p>        \u6b63\u786e<\/p>\n<p>        )<\/p>\n<p>       -\\\\log Q(\\\\text{\u6b63\u786e})<\/p>\n<p>    <\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u6b63\u786e<\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5e72\u51c0\u53ef\u7528\u3002<\/p>\n<p>\u65b9\u5411\u4e4b\u5dee\u7275\u51fa\u7684&#034;\u4e24\u4e2a KL \u8c01\u66f4\u5408\u9002&#034;&#xff0c;\u662f\u4e00\u7ec4\u66f4\u7ec6\u7684\u53d6\u820d&#xff0c;\u7559\u5230\u4e0b\u4e00\u7bc7\u5c55\u5f00\u3002<\/p>\n<hr \/>\n<h2>\u4e24\u4e2a\u5e38\u89c1\u7591\u95ee<\/h2>\n<ul>\n<li>\u201c\u4ea4\u53c9\u71b5\u4e00\u5b9a\u4e0d\u5c0f\u4e8e\u71b5\u5417&#xff1f;\u201d&#xff1a;\u662f\u3002\u7531 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>          K<\/p>\n<p>          L<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         \u2225<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>        H(P,Q)&#061;H(P)&#043;\\\\mathrm{KL}(P\\\\|Q)<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e14 KL \u6052\u975e\u8d1f&#xff0c;\u7acb\u523b\u5f97 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         ,<\/p>\n<p>         Q<\/p>\n<p>         )<\/p>\n<p>         \u2265<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         P<\/p>\n<p>         )<\/p>\n<p>        H(P,Q)\\\\ge H(P)<\/p>\n<p>     <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7b49\u53f7\u53ea\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>         &#061;<\/p>\n<p>         P<\/p>\n<p>        Q&#061;P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u65f6\u6210\u7acb\u3002\u4ea4\u53c9\u71b5\u7684\u4e0b\u754c\u5c31\u662f\u71b5&#xff0c;\u8fd9\u4e5f\u8bf4\u660e&#034;\u7528\u9519\u7684\u5206\u5e03\u63cf\u8ff0\u771f\u76f8&#xff0c;\u53ea\u4f1a\u591a\u4ed8\u4ee3\u4ef7\u3001\u4e0d\u4f1a\u66f4\u7701&#034;\u3002<\/li>\n<li>\u201c\u4ea4\u53c9\u71b5\u4f1a\u4e0d\u4f1a\u7b97\u6210\u8d1f\u6570&#xff1f;\u201d&#xff1a;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        Q(x)&gt;0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/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> \u65f6\u4e0d\u4f1a\u3002\u6bcf\u4e00\u9879 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2212<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>        -P(x)\\\\log Q(x)<\/p>\n<p>     <\/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=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u91cc <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2212<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u2265<\/p>\n<p>         0<\/p>\n<p>        -\\\\log Q(x)\\\\ge0<\/p>\n<p>     <\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/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>&#xff08;\u56e0\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u2264<\/p>\n<p>         1<\/p>\n<p>        Q(x)\\\\le1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/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\">1<\/span><\/span><\/span><\/span><\/span>&#xff09;\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         x<\/p>\n<p>         )<\/p>\n<p>         \u2265<\/p>\n<p>         0<\/p>\n<p>        P(x)\\\\ge0<\/p>\n<p>     <\/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=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/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>&#xff0c;\u52a0\u6743\u6c42\u548c\u81ea\u7136\u975e\u8d1f\u3002\u5b9e\u8df5\u4e2d\u7528 log-sum-exp \u7a33\u5b9a\u8ba1\u7b97&#xff08;\u7b2c 235 \u7bc7\u8bb2\u8fc7&#xff09;&#xff0c;\u4e5f\u4e0d\u4f1a\u56e0\u4e3a\u6570\u503c\u4e0b\u6ea2\u51fa\u8d1f\u3002<\/li>\n<\/ul>\n<hr \/>\n<h2>\u5c0f\u7ed3<\/h2>\n<p>\u4ea4\u53c9\u71b5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        \u2212<\/p>\n<p>         \u2211<\/p>\n<p>         x<\/p>\n<p>        P<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        Q<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>       H(P,Q)&#061;-\\\\sum_x P(x)\\\\log Q(x)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 1.0497em;vertical-align: -0.2997em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.0017em\"><span class=\"\" style=\"top: -2.4003em;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\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/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\">x<\/span><span class=\"mclose\">)<\/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\">Q<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u662f&#034;\u7528\u9884\u6d4b\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u53bb\u6c47\u62a5\u771f\u5b9e\u5206\u5e03 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u7684\u5e73\u5747\u610f\u5916&#034;&#xff0c;\u5206\u5de5\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>        P<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u5f53\u6743\u91cd\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         Q<\/p>\n<p>        Q<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u8fdb log\u3002\u6743\u91cd\u5fc5\u987b\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u56e0\u4e3a\u6837\u672c\u4ece\u771f\u5b9e\u5206\u5e03\u91c7\u51fa\u3001\u5e73\u5747\u503c\u53ea\u80fd\u6309 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u52a0\u6743&#xff1b;\u4ece\u6700\u5927\u4f3c\u7136\u51fa\u53d1\u7684\u63a8\u5bfc\u4f1a\u81ea\u7136\u628a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u9001\u5230\u6743\u91cd\u4f4d&#xff0c;\u5e76\u8bc1\u660e&#034;\u6700\u5927\u4f3c\u7136 &#061; \u6700\u5c0f\u5316\u4ea4\u53c9\u71b5&#034;\u3002\u62c6\u5f00\u540e\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        ,<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>         K<\/p>\n<p>         L<\/p>\n<p>        (<\/p>\n<p>        P<\/p>\n<p>        \u2225<\/p>\n<p>        Q<\/p>\n<p>        )<\/p>\n<p>       H(P,Q)&#061;H(P)&#043;\\\\mathrm{KL}(P\\\\|Q)<\/p>\n<p>    <\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">Q<\/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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4e8e\u662f\u6700\u5c0f\u5316\u4ea4\u53c9\u71b5\u5c31\u662f\u8ba9 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Q<\/p>\n<p>       Q<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8778em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">Q<\/span><\/span><\/span><\/span><\/span> \u903c\u8fd1 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>\u3002\u53cd\u8fc7\u6765\u5199 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        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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: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/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: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathrm\">KL<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">Q<\/span><span class=\"mord\">\u2225<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6570\u503c\u4e0d\u540c\u3001\u4e14\u5728 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