{"id":117443,"date":"2026-10-24T21:48:25","date_gmt":"2026-10-24T13:48:25","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/117443.html"},"modified":"2026-10-24T21:48:25","modified_gmt":"2026-10-24T13:48:25","slug":"%e3%80%8a%e4%bf%a1%e6%81%af%e8%ae%ba%e5%9f%ba%e7%a1%80%e3%80%8b%e7%ac%ac2%e7%ab%a0-%e7%a6%bb%e6%95%a3%e4%bf%a1%e6%81%af%e7%9a%84%e5%ba%a6%e9%87%8f-2-42-6-%e5%ad%a6%e4%b9%a0%e7%ac%94%e8%ae%b0","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/117443.html","title":{"rendered":"\u300a\u4fe1\u606f\u8bba\u57fa\u7840\u300b\u7b2c2\u7ae0 \u79bb\u6563\u4fe1\u606f\u7684\u5ea6\u91cf |2.4~2.6 \u5b66\u4e60\u7b14\u8bb0"},"content":{"rendered":"

\/* MD \/ \u5bcc\u6587\u672c\u4e2d\u7684 .toc\uff08\u542b\u535a\u5ba2\u56ed\u642c\u5bb6\u7b49\u5d4c\u5957\u7ed3\u6784\uff09\uff1b.toc-box \u5728\u4fa7\u680f\uff0c\u4e0d\u53d7\u5f71\u54cd *\/
\n #content_views .toc,
\n \/* \u7f16\u8f91\u5668\u5e38\u5728\u76ee\u5f55\u524d\u540e\u63d2\u5165\u7a7a p\uff08:empty \u4ecd\u5360 20px\uff09\uff0c\u4e00\u5e76\u53bb\u6389\u907f\u514d\u9876\u7a7a\u9699 *\/
\n #content_views.markdown_views > p:empty:has(+ .toc),
\n #content_views.markdown_views > .toc + p:empty,
\n \/* \u5bcc\u6587\u672c\u65e7\u7248\u76ee\u5f55\u6807\u8bb0 *\/
\n #content_views.htmledit_views #main-toc,
\n #content_views.htmledit_views #hr-toc,
\n #content_views.htmledit_views p[id*=\\”-toc\\”] {
\n display: none !important;
\n }
\n \/* \u76ee\u5f55\u53bb\u6389\u540e\uff0c\u7d27\u8ddf\u7684\u9996\u4e2a\u6807\u9898\u4e0d\u518d\u591a\u51fa\u4e00\u5757\u4e0a\u8fb9\u8ddd *\/
\n #content_views.markdown_views > .toc + h1,
\n #content_views.markdown_views > .toc + h2,
\n #content_views.markdown_views > .toc + h3,
\n #content_views.markdown_views > .toc + h4,
\n #content_views.markdown_views > .toc + p:empty + h1,
\n #content_views.markdown_views > .toc + p:empty + h2,
\n #content_views.markdown_views > .toc + p:empty + h3,
\n #content_views.markdown_views > .toc + p:empty + h4 {
\n margin-top: 0 !important;
\n }<\/p>\n

1. \u5f15\u8a00<\/h3>\n

\u672c\u7ae0\u56f4\u7ed5\u79bb\u6563\u4fe1\u6e90\u7684\u71b5\u3001\u8054\u5408\u71b5\u3001\u6761\u4ef6\u71b5\u4e0e\u4e92\u4fe1\u606f\u5c55\u5f00，\u662f\u4fe1\u606f\u8bba\u7684\u6838\u5fc3\u5ea6\u91cf\u4f53\u7cfb\u30022.4~2.6 \u8282\u5728 2.1~2.3 \u8282\u5b9a\u4e49\u71b5\u7684\u57fa\u7840\u4e0a，\u8fdb\u4e00\u6b65\u5f15\u5165\u8054\u5408\u71b5\u3001\u6761\u4ef6\u71b5\u3001\u4e92\u4fe1\u606f\u53ca\u5176\u94fe\u5f0f\u6cd5\u5219，\u5e76\u8ba8\u8bba\u71b5\u51fd\u6570\u7684\u6027\u8d28\u4e0e\u5178\u578b\u5e8f\u5217\u3002\u672c\u8282\u7b14\u8bb0\u4ee5\u6982\u5ff5\u68b3\u7406 + \u516c\u5f0f\u63a8\u5bfc + \u4f8b\u9898\u89e3\u6790\u7684\u65b9\u5f0f\u5448\u73b0，\u5e2e\u52a9\u8bfb\u8005\u5efa\u7acb\u4ece"\u5355\u4e2a\u7b26\u53f7\u7684\u71b5"\u5230"\u591a\u4e2a\u7b26\u53f7\u8054\u5408\u5ea6\u91cf"\u7684\u5b8c\u6574\u8ba4\u77e5\u3002<\/p>\n

2. 2.4 \u8054\u5408\u71b5\u4e0e\u6761\u4ef6\u71b5<\/h3>\n

2.4.1 \u8054\u5408\u71b5\u7684\u5b9a\u4e49<\/h4>\n

\u8bbe\u79bb\u6563\u968f\u673a\u53d8\u91cf XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e0e YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u4e3a p(x,y)p(x,y)<\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5219\u8054\u5408\u71b5\u5b9a\u4e49\u4e3a：<\/p>\n

H(X,Y)=\u2212\u2211x\u2208X\u2211y\u2208Yp(x,y)log\u2061p(x,y)
\nH(X,Y) = -\\\\sum_{x \\\\in \\\\mathcal{X}} \\\\sum_{y \\\\in \\\\mathcal{Y}} p(x,y) \\\\log p(x,y)
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>x<\/span>\u2208<\/span>X<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span>\u2208<\/span>Y<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span>log<\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8054\u5408\u71b5\u5ea6\u91cf\u7684\u662f\u540c\u65f6\u89c2\u6d4b XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e0e YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u65f6\u6240\u9700\u7684\u5e73\u5747\u4fe1\u606f\u91cf\u3002\u5f53 XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e0e YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u72ec\u7acb\u65f6，H(X,Y)=H(X)+H(Y)H(X,Y) = H(X) + H(Y)<\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span>；\u5426\u5219\u8054\u5408\u71b5\u5c0f\u4e8e\u4e8c\u8005\u4e4b\u548c\u3002<\/p>\n

2.4.2 \u6761\u4ef6\u71b5\u7684\u5b9a\u4e49<\/h4>\n

\u5df2\u77e5 YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u7684\u6761\u4ef6\u4e0b，XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u6761\u4ef6\u71b5\u5b9a\u4e49\u4e3a：<\/p>\n

H(X\u2223Y)=\u2211y\u2208Yp(y)H(X\u2223Y=y)=\u2212\u2211x,yp(x,y)log\u2061p(x\u2223y)
\nH(X|Y) = \\\\sum_{y \\\\in \\\\mathcal{Y}} p(y) H(X|Y=y) = -\\\\sum_{x,y} p(x,y) \\\\log p(x|y)
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>y<\/span>\u2208<\/span>Y<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>y<\/span>)<\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span><\/span>=<\/span><\/span><\/span><\/span>y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>x<\/span>,<\/span>y<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span>log<\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>\u2223<\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6761\u4ef6\u71b5\u5ea6\u91cf\u7684\u662f\u5728\u5df2\u77e5 YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u4e4b\u540e，XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4ecd\u5269\u4f59\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u76f4\u89c2\u7406\u89e3：\u89c2\u6d4b\u5230 YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u4f1a\u6d88\u9664\u4e00\u90e8\u5206\u5173\u4e8e XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u786e\u5b9a\u6027，\u5269\u4f59\u90e8\u5206\u5373\u4e3a H(X\u2223Y)H(X|Y)<\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

2.4.3 \u94fe\u5f0f\u6cd5\u5219<\/h4>\n

\u8054\u5408\u71b5\u4e0e\u6761\u4ef6\u71b5\u4e4b\u95f4\u6ee1\u8db3\u94fe\u5f0f\u6cd5\u5219：<\/p>\n

H(X,Y)=H(X)+H(Y\u2223X)=H(Y)+H(X\u2223Y)
\nH(X,Y) = H(X) + H(Y|X) = H(Y) + H(X|Y)
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>\u2223<\/span>X<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u63a8\u5e7f\u5230 nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u4e2a\u53d8\u91cf：<\/p>\n

H(X1,X2,\u2026,Xn)=\u2211i=1nH(Xi\u2223Xi\u22121,\u2026,X1)
\nH(X_1, X_2, \\\\dots, X_n) = \\\\sum_{i=1}^{n} H(X_i | X_{i-1}, \\\\dots, X_1)
\n<\/span><\/span>H<\/span>(<\/span>X<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>X<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>X<\/span><\/span>i<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>X<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f8b\u9898：\u8bbe XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u4e0e YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u7684\u8054\u5408\u5206\u5e03\u4e3a p(0,0)=1\/2p(0,0)=1\/2<\/span><\/span>p<\/span>(<\/span>0<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/2<\/span><\/span><\/span><\/span><\/span>，p(0,1)=1\/4p(0,1)=1\/4<\/span><\/span>p<\/span>(<\/span>0<\/span>,<\/span><\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/4<\/span><\/span><\/span><\/span><\/span>，p(1,0)=0p(1,0)=0<\/span><\/span>p<\/span>(<\/span>1<\/span>,<\/span><\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，p(1,1)=1\/4p(1,1)=1\/4<\/span><\/span>p<\/span>(<\/span>1<\/span>,<\/span><\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/4<\/span><\/span><\/span><\/span><\/span>\u3002\u6c42 H(X)H(X)<\/span><\/span>H<\/span>(<\/span>X<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3001H(Y)H(Y)<\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3001H(X\u2223Y)H(X|Y)<\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e0e H(X,Y)H(X,Y)<\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u89e3：\u5148\u6c42\u8fb9\u7f18\u5206\u5e03：<\/p>\n

pX(0)=1\/2+1\/4=3\/4,pX(1)=1\/4
\np_X(0) = 1\/2 + 1\/4 = 3\/4, \\\\quad p_X(1) = 1\/4
\n<\/span><\/span>p<\/span><\/span>X<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/2<\/span><\/span>+<\/span><\/span><\/span><\/span>1\/4<\/span><\/span>=<\/span><\/span><\/span><\/span>3\/4<\/span>,<\/span><\/span><\/span>p<\/span><\/span>X<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/4<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

pY(0)=1\/2+0=1\/2,pY(1)=1\/4+1\/4=1\/2
\np_Y(0) = 1\/2 + 0 = 1\/2, \\\\quad p_Y(1) = 1\/4 + 1\/4 = 1\/2
\n<\/span><\/span>p<\/span><\/span>Y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/2<\/span><\/span>+<\/span><\/span><\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/2<\/span>,<\/span><\/span><\/span>p<\/span><\/span>Y<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/4<\/span><\/span>+<\/span><\/span><\/span><\/span>1\/4<\/span><\/span>=<\/span><\/span><\/span><\/span>1\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e8e\u662f：<\/p>\n

H(X)=\u221234log\u206134\u221214log\u206114\u22480.811\u00a0bit
\nH(X) = -\\\\frac{3}{4}\\\\log\\\\frac{3}{4} – \\\\frac{1}{4}\\\\log\\\\frac{1}{4} \\\\approx 0.811 \\\\text{ bit}
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>0.811<\/span>\u00a0bit<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

H(Y)=\u221212log\u206112\u221212log\u206112=1\u00a0bit
\nH(Y) = -\\\\frac{1}{2}\\\\log\\\\frac{1}{2} – \\\\frac{1}{2}\\\\log\\\\frac{1}{2} = 1 \\\\text{ bit}
\n<\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>\u00a0bit<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8054\u5408\u71b5：<\/p>\n

H(X,Y)=\u221212log\u206112\u221214log\u206114\u221214log\u206114=1.5\u00a0bit
\nH(X,Y) = -\\\\frac{1}{2}\\\\log\\\\frac{1}{2} – \\\\frac{1}{4}\\\\log\\\\frac{1}{4} – \\\\frac{1}{4}\\\\log\\\\frac{1}{4} = 1.5 \\\\text{ bit}
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1.5<\/span>\u00a0bit<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u94fe\u5f0f\u6cd5\u5219：<\/p>\n

H(X\u2223Y)=H(X,Y)\u2212H(Y)=1.5\u22121=0.5\u00a0bit
\nH(X|Y) = H(X,Y) – H(Y) = 1.5 – 1 = 0.5 \\\\text{ bit}
\n<\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>,<\/span><\/span>Y<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1.5<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span>=<\/span><\/span><\/span><\/span>0.5<\/span>\u00a0bit<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

3. 2.5 \u4e92\u4fe1\u606f<\/h3>\n

3.1 \u4e92\u4fe1\u606f\u7684\u5b9a\u4e49<\/h4>\n

\u4e92\u4fe1\u606f\u5ea6\u91cf\u7684\u662f\u89c2\u6d4b YY<\/span><\/span>Y<\/span><\/span><\/span><\/span><\/span> \u4e4b\u540e，\u5173\u4e8e XX<\/span><\/span>X<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\u91cf：<\/p>\n

I(X;Y)=H(X)\u2212H(X\u2223Y)=H(Y)\u2212H(Y\u2223X)
\nI(X;Y) = H(X) – H(X|Y) = H(Y) – H(Y|X)
\n<\/span><\/span>I<\/span>(<\/span>X<\/span>;<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>H<\/span>(<\/span>X<\/span>\u2223<\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>)<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>H<\/span>(<\/span>Y<\/span>\u2223<\/span>X<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7b49\u4ef7\u5730，\u7528\u8054\u5408\u5206\u5e03\u4e0e\u8fb9\u7f18\u5206\u5e03\u7684 KL \u6563\u5ea6\u8868\u793a：<\/p>\n

I(X;Y)=\u2211x,yp(x,y)log\u2061p(x,y)p(x)p(y)
\nI(X;Y) = \\\\sum_{x,y} p(x,y) \\\\log \\\\frac{p(x,y)}{p(x)p(y)}
\n<\/span><\/span>I<\/span>(<\/span>X<\/span>;<\/span><\/span>Y<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>x<\/span>,<\/span>y<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span>log<\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>)<\/span>p<\/span>(<\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

3.2 \u4e92\u4fe1\u606f\u7684\u6027\u8d28<\/h4>\n