![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729193638837.png\")
Deep Learning Math<\/h1>\n\n
Deep Learning Math<\/h1>\n\u4fe1\u606f\u8bba\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u81f3\u5173\u91cd\u8981\u3002\u4fe1\u606f\u91cf\u548c\u71b5\u5ea6\u91cf\u6570\u636e\u7684\u4e0d\u786e\u5b9a\u6027\u548c\u4fe1\u606f\u91cf\u3002\u76f8\u5bf9\u71b5\uff08Kullback-Leibler \u6563\u5ea6\uff09\u548c\u4ea4\u53c9\u71b5\u7528\u4e8e\u8861\u91cf\u6982\u7387\u5206\u5e03\u7684\u5dee\u5f02\uff0c\u4ea4\u53c9\u71b5\u5e38\u7528\u4e8e\u5206\u7c7b\u4efb\u52a1\u4e2d\u7684\u635f\u5931\u51fd\u6570\u3002\u4e92\u4fe1\u606f\u5219\u7528\u4e8e\u5ea6\u91cf\u4e24\u4e2a\u53d8\u91cf\u95f4\u7684\u5171\u4eab\u4fe1\u606f\u91cf\uff0c\u5e2e\u52a9\u7406\u89e3\u53d8\u91cf\u4f9d\u8d56\u6027\uff0c\u5e38\u7528\u4e8e\u7279\u5f81\u9009\u62e9\u3002<\/p>\n
Jensen \u4e0d\u7b49\u5f0f\u3001Chebyshev \u4e0d\u7b49\u5f0f\u548c Pinsker \u4e0d\u7b49\u5f0f\u5728\u4fe1\u606f\u8bba\u4e2d\u7528\u4e8e\u4f30\u8ba1\u548c\u754c\u5b9a\u6982\u7387\u5206\u5e03\u7684\u6027\u8d28\u3002\u901a\u8fc7\u8fd9\u4e9b\u5de5\u5177\uff0c\u6df1\u5ea6\u5b66\u4e60\u80fd\u591f\u4f18\u5316\u6a21\u578b\u6027\u80fd\uff0c\u63d0\u5347\u9884\u6d4b\u51c6\u786e\u6027\u3002<\/p>\n
![](\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\")
Agenda<\/h3>\n\u4fe1\u606f\u91cf\u548c\u71b5<\/p>\n
\u4e92\u4fe1\u606f<\/p>\n
\u4e0d\u7b49\u5f0f<\/p>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729193701124.png\")
Additional Packages for Google Colab<\/h4>\nIf you are using Google Colab<\/a>, you have to install additional packages. To do this, simply run the following cell.<\/p>\n \u5f53\u4e00\u4e2a\u4e8b\u4ef6\u53d1\u751f\u65f6\uff0c\u6211\u4eec\u4f1a\u7ecf\u5386\u4ece\u4e0d\u77e5\u9053\u8be5\u4e8b\u4ef6\u662f\u5426\u4f1a\u53d1\u751f\u5230\u77e5\u9053\u5b83\u5df2\u7ecf\u53d1\u751f\u7684\u53d8\u5316\u3002\u5728\u8fd9\u4e00\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u7684\u8ba4\u77e5\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\u4e86\uff0c\u8fd9\u79cd\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\u7684\u91cf\u5373\u4e3a\u4fe1\u606f\u91cf\u3002\u4fe1\u606f\u91cf\u8d8a\u5927\uff0c\u8868\u793a\u8be5\u4e8b\u4ef6\u53d1\u751f\u6240\u5e26\u6765\u7684\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\u8d8a\u591a\u3002\u901a\u5e38\uff0c\u4fe1\u606f\u91cf\u7528\u6bd4\u7279\uff08bit\uff09\u6765\u8868\u793a\u3002<\/p>\n \u5bf9\u4e8e\u67d0\u4e00\u4e2a\u79bb\u6563\u4e8b\u4ef6 $x$ \uff0c\u5176\u4fe1\u606f\u91cf $I(x)$ \u53ef\u4ee5\u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n \u6bd4\u7279\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u7684\u57fa\u672c\u5355\u4f4d\u3002\u4e00\u4e2a\u6bd4\u7279\u7684\u4fe1\u606f\u91cf\u8868\u793a\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d ( 0 \u6216 1) \u7684\u4fe1\u606f\u3002\u5728\u9009\u62e9\u5bf9\u6570\u7684\u5e95\u4e3a 2 \u7684\u60c5\u51b5\u4e0b\uff0c\u5982\u679c\u4e00\u4e2a\u4e8b\u4ef6\u7684\u53d1\u751f\u6982\u7387 $P(x)$ \u4e3a 0.5 \uff0c\u90a3\u4e48\u5176\u4fe1\u606f\u91cf\u4e3a: \u7eb3\u7279\u662f\u4fe1\u606f\u91cf\u7684\u81ea\u7136\u5355\u4f4d\uff0c\u5728\u5e95\u6570\u53d6\u81ea\u7136\u5bf9\u6570 $e$ \u65f6\u4f7f\u7528\u3002\u4f8b\u5982\uff0c\u5982\u679c\u4e00\u4e2a\u4e8b\u4ef6\u7684\u53d1\u751f\u6982\u7387 $P(x)$ \u4e3a 0.5 \uff0c\u90a3\u4e48\u5176\u4fe1\u606f\u91cf\u4e3a: \u4fe1\u606f\u91cf\u8868\u793a\u5f53\u4e8b\u4ef6 $x$ \u53d1\u751f\u65f6\uff0c\u6211\u4eec\u5bf9\u8be5\u4e8b\u4ef6\u53d1\u751f\u7684\u786e\u5b9a\u6027\u589e\u52a0\u4e86\u591a\u5c11\u3002\u4e0d\u786e\u5b9a\u6027\u8d8a\u5927\uff08\u5373\u4e8b\u4ef6\u8d8a\u4e0d\u5e38\u89c1\uff09\uff0c\u5176\u4fe1\u606f\u91cf\u8d8a\u5927\u3002<\/p>\n \u71b5\uff08Entropy\uff09\u662f\u4fe1\u606f\u8bba\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c\u7528\u6765\u8861\u91cf\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u71b5\u7684\u6982\u5ff5\u6700\u65e9\u7531\u514b\u52b3\u5fb7\u00b7\u9999\u519c\uff08Claude Shannon\uff09\u5728\u5176\u5f00\u521b\u6027\u7684\u8bba\u6587\u300a\u901a\u4fe1\u7684\u6570\u5b66\u7406\u8bba\u300b\uff08A Mathematical Theory of Communication\uff09\u4e2d\u63d0\u51fa\u3002\u71b5\u5728\u4fe1\u606f\u8bba\u3001\u7edf\u8ba1\u5b66\u3001\u70ed\u529b\u5b66\u7b49\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u548c\u6df1\u8fdc\u7684\u5f71\u54cd\u3002<\/p>\n \u71b5\u7684\u5b9a\u4e49\uff1a<\/p>\n \u5bf9\u4e8e\u4e00\u4e2a\u79bb\u6563\u968f\u673a\u53d8\u91cf $X$ \u53ca\u5176\u6982\u7387\u5206\u5e03 $P(X)$ \uff0c\u6458 $H(X)$ \u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n \u71b5\u8868\u793a\u5728\u957f\u65f6\u95f4\u89c2\u5bdf\u8be5\u968f\u673a\u53d8\u91cf\u65f6\uff0c\u6211\u4eec\u6bcf\u6b21\u89c2\u6d4b\u6240\u83b7\u5f97\u7684\u4fe1\u606f\u91cf\u7684\u5e73\u5747\u503c\u3002\u5b83\u53cd\u6620\u4e86\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u4e0d\u786e\u5b9a\u6027\u7a0b\u5ea6\u3002<\/p>\n \u4fe1\u606f\u91cf\u548c\u71b5\u7684\u57fa\u672c\u8054\u7cfb\u5728\u4e8e\u5b83\u4eec\u90fd\u662f\u57fa\u4e8e\u4e8b\u4ef6\u53d1\u751f\u6982\u7387 $P(x)$ \u7684\u5bf9\u6570\u8ba1\u7b97\u7684\u5ea6\u91cf\u3002\u4fe1\u606f\u91cf\u8861\u91cf\u5355\u4e2a\u4e8b\u4ef6\u5e26\u6765\u7684\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\uff0c\u800c\u71b5\u662f\u6240\u6709\u53ef\u80fd\u4e8b\u4ef6\u7684\u4fe1\u606f\u91cf\u7684\u52a0\u6743\u5e73\u5747\u503c\u3002<\/strong><\/p>\n \u71b5\u53ef\u4ee5\u770b\u4f5c\u662f\u4fe1\u606f\u91cf\u7684\u671f\u671b\u503c<\/strong>\u3002\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u53d8\u91cf $X$ \uff0c\u71b5 $H(X)$ \u662f\u5176\u53ef\u80fd\u53d6\u503c\u7684\u4fe1\u606f\u91cf $I(x)$ \u7684\u671f\u671b, \u5373: \u8fd9\u610f\u5473\u7740\u71b5\u662f\u6bcf\u4e2a\u4e8b\u4ef6\u7684\u4fe1\u606f\u91cf\u4e58\u4ee5\u5176\u53d1\u751f\u6982\u7387\u7684\u603b\u548c\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u71b5\u8861\u91cf\u4e86\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u6574\u4f53\u4e0d\u786e\u5b9a\u6027\uff0c\u800c\u4e0d\u662f\u5355\u4e2a\u4e8b\u4ef6\u7684\u4fe1\u606f\u91cf\u3002<\/p>\n \u71b5\u7684\u6027\u8d28<\/strong>\uff1a<\/p>\n \u6458\u603b\u662f\u975e\u8d1f\u7684\uff0c\u5373 $H(X) \\geq 0$ \u3002\u8fd9\u662f\u56e0\u4e3a\u6982\u7387 $P(x)$ \u603b\u662f\u5927\u4e8e 0 \u4e14\u5c0f\u4e8e\u7b49\u4e8e 1 \uff0c\u56e0\u800c\u5bf9\u6570\u503c $\\log _b P(x)$ \u603b\u662f\u5c0f\u4e8e\u7b49\u4e8e 0 \u3002\u7531\u4e8e\u71b5\u662f\u8fd9\u4e9b\u503c\u7684\u52a0\u6743\u548c\u7684\u8d1f\u503c\uff0c\u6240\u4ee5\u6458\u603b\u662f\u975e\u8d1f\u7684\u3002<\/p>\n \u5982\u679c\u968f\u673a\u53d8\u91cf $X$ \u662f\u786e\u5b9a\u7684\uff08\u5373\u5176\u6982\u7387\u5206\u5e03\u4e3a\u67d0\u4e00\u4e2a\u7279\u5b9a\u503c\u7684\u6982\u7387\u4e3a 1 \uff0c\u5176\u4ed6\u4e3a 0 \uff09\uff0c\u5219\u4431 $H(X)=0$ \u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0d\u786e\u5b9a\u6027\u5b8c\u5168\u6d88\u9664\uff0c\u6ca1\u6709\u4fe1\u606f\u9700\u8981\u4f20\u9012\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u786e\u5b9a\u7684\u4e8b\u4ef6\uff0c\u5176\u6982\u7387 $P(x)=1$ \uff0c\u5219: \u5bf9\u4e8e\u4e00\u4e2a\u6709 $n$ \u4e2a\u53ef\u80fd\u503c\u4e14\u6bcf\u4e2a\u503c\u6982\u7387\u5747\u7b49\u7684\u79bb\u6563\u968f\u673a\u53d8\u91cf $X$ \uff0c\u6458\u8fbe\u5230\u6700\u5927\u503c $\\log _b n$ \u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf $X$ \uff0c\u5176\u6bcf\u4e2a\u53ef\u80fd\u503c\u7684\u6982\u7387\u5747\u4e3a $\\frac{1}{n}$ \uff0c\u5219\u6458\u8ba1\u7b97\u5982\u4e0b: \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u7531\u4e8e\u6bcf\u4e2a\u53ef\u80fd\u503c\u7684\u6982\u7387\u5747\u7b49\uff0c\u4e0d\u786e\u5b9a\u6027\u8fbe\u5230\u6700\u5927\uff0c\u56e0\u6b64\u6458\u4e5f\u8fbe\u5230\u6700\u5927\u503c\u3002<\/p>\n \u603b\u7684\u6765\u8bf4\uff0c\u71b5\u662f\u8861\u91cf\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u4e0d\u786e\u5b9a\u6027\u7684\u5ea6\u91cf\uff0c\u662f\u4fe1\u606f\u8bba\u4e2d\u7684\u6838\u5fc3\u6982\u5ff5\u3002\u901a\u8fc7\u5b9a\u4e49\u548c\u6027\u8d28\u7684\u7406\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u638c\u63e1\u4fe1\u606f\u71b5\u7684\u672c\u8d28\uff0c\u5e76\u5728\u6570\u636e\u5206\u6790\u3001\u901a\u4fe1\u3001\u7f16\u7801\u3001\u673a\u5668\u5b66\u4e60\u7b49\u9886\u57df\u4e2d\u5e94\u7528\u8fd9\u4e00\u6982\u5ff5\u6765\u4f18\u5316\u4fe1\u606f\u5904\u7406\u548c\u4f20\u8f93\u7684\u6548\u7387\u3002\u71b5\u7684\u975e\u8d1f\u6027\u3001\u786e\u5b9a\u6027\u548c\u6700\u5927\u503c\u6027\u8d28\u4f7f\u5176\u5728\u5404\u79cd\u573a\u666f\u4e0b\u90fd\u6709\u91cd\u8981\u7684\u7406\u8bba\u548c\u5b9e\u9645\u610f\u4e49\u3002<\/p>\n \u8054\u5408\u71b5 $H(X, Y)$ \u8861\u91cf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \u7684\u8054\u5408\u4e0d\u786e\u5b9a\u6027\uff0c\u5b83\u6269\u5c55\u4e86\u5355\u4e2a\u968f\u673a\u53d8\u91cf\u7684\u71b5\u7684\u6982\u5ff5\uff0c\u901a\u8fc7\u8003\u8651\u4e24\u4e2a\u53d8\u91cf\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u6765\u8bc4\u4f30\u7cfb\u7edf\u7684\u6574\u4f53\u4e0d\u786e\u5b9a\u6027\u3002\u5b9a\u4e49\u4e3a: \u5176\u4e2d\uff1a<\/p>\n \u8054\u5408\u6458\u8861\u91cf\u7684\u662f\u5728\u540c\u65f6\u89c2\u5bdf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \u65f6\u7cfb\u7edf\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u5b83\u8003\u8651\u4e86\u8fd9\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u548c\u76f8\u4e92\u4f9d\u8d56\u6027\uff0c\u56e0\u6b64\u6bd4\u5355\u4e2a\u53d8\u91cf\u7684\u6458\u63d0\u4f9b\u4e86\u66f4\u591a\u7684\u4fe1\u606f\u3002<\/p>\n \u60c5\u51b5\u4e00: \u72ec\u7acb\u968f\u673a\u53d8\u91cf<\/strong><\/p>\n \u5982\u679c $X$ \u548c $Y$ \u662f\u72ec\u7acb\u7684\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48\u5b83\u4eec\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\u5355\u53d8\u91cf\u6982\u7387\u5206\u5e03\u7684\u4e58\u79ef\uff0c\u5373 $P(x, y)=P(x) P(y)$ \u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8054\u5408\u71b5 $H(X, Y)$ \u53ef\u4ee5\u5206\u89e3\u4e3a\u4e24\u4e2a\u5355\u53d8\u91cf\u71b5\u7684\u548c\uff1a \u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u5f7c\u6b64\u72ec\u7acb\uff0c\u5b83\u4eec\u7684\u8054\u5408\u4e0d\u786e\u5b9a\u6027\u7b49\u4e8e\u5404\u81ea\u4e0d\u786e\u5b9a\u6027\u7684\u603b\u548c\u3002<\/p>\n \u60c5\u51b5\u4e8c: \u76f8\u5173\u968f\u673a\u53d8\u91cf<\/strong> \u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u4f7f\u7528\u8054\u5408\u6982\u7387\u5206\u5e03\u6765\u76f4\u63a5\u8ba1\u7b97\u8054\u5408\u71b5 $H(X, Y)$ \u3002\u8fd9\u662f\u4e00\u4e2a\u5305\u542b\u6240\u6709\u53ef\u80fd\u7684 $(x,y)$ \u7ec4\u5408\u53ca\u5176\u5bf9\u5e94\u6982\u7387\u7684\u5206\u5e03\u8868\u3002<\/p>\n \u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u76f8\u5173\u7684\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \uff0c\u5b83\u4eec\u7684\u8054\u5408\u6982\u7387\u5206\u5e03\u5982\u4e0b\u8868\u6240\u793a:<\/p>\n \n \u6211\u4eec\u9009\u62e9\u5bf9\u6570\u7684\u5e95 $b=2$ \uff0c\u5219\u8054\u5408\u6458 $H(X, Y)$ \u8ba1\u7b97\u5982\u4e0b:<\/p>\n \u8ba1\u7b97\u6bcf\u4e2a $(x, y)$ \u7ec4\u5408\u7684\u8d21\u732e:<\/p>\n \u7d2f\u52a0\u6240\u6709\u7ec4\u5408\u7684\u8d21\u732e: \u6761\u4ef6\u71b5 $H(Y \\mid X)$ \u7528\u6765\u8861\u91cf\u5728\u5df2\u77e5\u968f\u673a\u53d8\u91cf $X$ \u7684\u60c5\u51b5\u4e0b\uff0c\u968f\u673a\u53d8\u91cf $Y$ \u7684\u5269\u4f59\u4e0d\u786e\u5b9a\u6027\u3002\u901a\u8fc7\u6761\u4ef6\u71b5\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u6df1\u5165\u5730\u7406\u89e3\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\u53ca\u5176\u4f9d\u8d56\u6027\u3002<\/p>\n \u6761\u4ef6\u71b5 $H(Y \\mid X)$ \u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n \u6761\u4ef6\u71b5 $H(Y \\mid X)$ \u8868\u793a\u5728\u5df2\u77e5 $X$ \u7684\u60c5\u51b5\u4e0b\uff0c $Y$ \u7684\u4e0d\u786e\u5b9a\u6027\u3002\u5b83\u8861\u91cf\u7684\u662f\u6211\u4eec\u5728\u77e5\u9053 $X$ \u4e4b\u540e\uff0c\u5bf9 $Y$ \u8fd8\u9700\u8981\u4e86\u89e3\u591a\u5c11\u4fe1\u606f\u3002<\/p>\n \u4e3e\u4e2a \u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \uff0c\u4f8b\u5982\u5929\u6c14\u72b6\u51b5 $X$ (\u6674\u5929\u3001\u96e8\u5929) \u548c\u4e0a\u73ed\u4ea4\u901a\u5de5\u5177\u9009\u62e9 $Y$ (\u5f00\u8f66\u3001\u5750\u516c\u4ea4)\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6761\u4ef6\u71b5 $H(Y \\mid X)$ \u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u5728\u77e5\u9053\u5929\u6c14\u72b6\u5144\u7684\u60c5\u51b5\u4e0b\uff0c\u5bf9\u4ea4\u901a\u5de5\u5177\u9009\u62e9\u7684\u5269\u4f59\u4e0d\u786e\u5b9a\u6027\u3002\u4f8b\u5982\uff0c\u5982\u679c\u5728\u6674\u5929\u548c\u96e8\u5929\u7684\u60c5\u51b5\u4e0b\uff0c\u4eba\u4eec\u9009\u62e9\u4ea4\u901a\u5de5\u5177\u7684\u884c\u4e3a\u6709\u663e\u8457\u5dee\u5f02\uff0c\u90a3\u4e48\u6761\u4ef6\u71b5\u4f1a\u8f83\u4f4e\uff0c\u8868\u660e\u5929\u6c14\u72b6\u51b5\u5bf9\u4ea4\u901a\u5de5\u5177\u9009\u62e9\u6709\u5f88\u597d\u7684\u9884\u6d4b\u6027\u3002<\/p>\n \u6761\u4ef6\u71b5\u7684\u6027\u8d28<\/strong><\/p>\n \u6761\u4ef6\u71b5\u7684\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\u662f\u94fe\u5f0f\u6cd5\u5219\uff0c\u5b83\u63cf\u8ff0\u4e86\u8054\u5408\u71b5\u3001\u5355\u53d8\u91cf\u71b5\u548c\u6761\u4ef6\u71b5\u4e4b\u95f4\u7684\u5173\u7cfb: \u8fd9\u610f\u5473\u7740\u8054\u5408\u71b5 $H(X, Y)$ \u53ef\u4ee5\u5206\u89e3\u4e3a $X$ \u7684\u71b5 $H(X)$ \u548c\u5728\u5df2\u77e5 $X$ \u7684\u60c5\u51b5\u4e0b $Y$ \u7684\u6761\u4ef6\u71b5 $H(Y \\mid$ $X)$ \u3002\u8fd9\u6761\u6cd5\u5219\u8868\u660e\u4e86\u4fe1\u606f\u7684\u4f20\u9012\u8fc7\u7a0b\uff0c\u5176\u4e2d $H(X)$ \u662f\u5173\u4e8e $X$ \u7684\u4fe1\u606f\uff0c\u800c $H(Y \\mid X)$ \u662f\u5728\u77e5\u9053 $X$\u540e\u9700\u8981\u989d\u5916\u83b7\u53d6\u7684\u5173\u4e8e $Y$ \u7684\u4fe1\u606f\u3002<\/p>\n \u6761\u4ef6\u71b5\u5177\u6709\u975e\u8d1f\u6027\uff0c\u5373 $H(Y \\mid X) \\geq 0$ \u3002\u8fd9\u662f\u56e0\u4e3a\u6761\u4ef6\u71b5\u8868\u793a\u7684\u662f\u4e0d\u786e\u5b9a\u6027\u7684\u5269\u4f59\u91cf\uff0c\u4e0d\u53ef\u80fd\u4e3a\u8d1f\u3002\u82e5 $H(Y \\mid X)=0$ \uff0c\u5219\u8bf4\u660e\u5728\u5df2\u77e5 $X$ \u7684\u60c5\u51b5\u4e0b\uff0c $Y$ \u5b8c\u5168\u786e\u5b9a\uff0c\u6ca1\u6709\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n \u76f8\u5bf9\u71b5\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03 $P, Q$ \u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u5b83\u901a\u8fc7\u8ba1\u7b97\u4e24\u4e2a\u5206\u5e03\u4e4b\u95f4\u7684\u4fe1\u606f\u5dee\u5f02\u6765\u8bc4\u4f30\u4e00\u4e2a\u5206\u5e03\u5728\u591a\u5927\u7a0b\u5ea6\u4e0a\u504f\u79bb\u4e86\u53e6\u4e00\u4e2a\u5206\u5e03\u3002<\/p>\n \u5bf9\u4e8e\u79bb\u6563\u6982\u7387\u5206\u5e03 $P$ \u548c $Q, \\mathrm{KL}$ \u6563\u5ea6\u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n KL \u6563\u5ea6\u5ea6\u91cf\u7684\u662f\u4f7f\u7528\u5206\u5e03 $Q$ \u6765\u8fd1\u4f3c\u5206\u5e03 $P$ \u65f6\u6240\u5f15\u5165\u7684\u4fe1\u606f\u635f\u5931\u3002\u5b83\u53cd\u6620\u4e86\u5728\u4f7f\u7528 $Q$ \u4ee3\u66ff $P$ \u7684\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u989d\u5916\u4e86\u89e3\u7684\u4fe1\u606f\u91cf\u3002<\/p>\n KL \u6563\u5ea6\u7684\u6027\u8d28<\/strong><\/p>\n KL \u6563\u5ea6\u5177\u6709\u975e\u8d1f\u6027\uff0c\u5373: \u4e14\u53ea\u6709\u5f53 $P=Q$ \u65f6\uff0c KL \u6563\u5ea6\u624d\u7b49\u4e8e\u96f6\u3002\u8fd9\u610f\u5473\u7740\u4e24\u4e2a\u5206\u5e03\u5b8c\u5168\u76f8\u540c\u65f6\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u5dee\u5f02\u4e3a\u96f6; \u800c\u5f53\u4e24\u4e2a\u5206\u5e03\u4e0d\u540c\u65f6\u65f6\uff0cKL \u6563\u5ea6\u59cb\u7ec8\u4e3a\u6b63\u503c\u3002<\/p>\n KL \u6563\u5ea6\u662f\u975e\u5bf9\u79f0\u7684\uff0c\u5373: \u8fd9\u610f\u5473\u7740\u4ece $P$ \u5230 $Q$ \u7684\u6563\u5ea6\u4e0e\u4ece $Q$ \u5230 $P$ \u7684\u6563\u5ea6\u662f\u4e0d\u540c\u7684\u3002\u56e0\u6b64\uff0cKL \u6563\u5ea6\u4e0d\u80fd\u7528\u6765\u8861\u91cf\u4e24\u4e2a\u5206\u5e03\u4e4b\u95f4\u7684\u5bf9\u79f0\u5dee\u5f02\uff0c\u800c\u662f\u66f4\u9002\u5408\u8861\u91cf\u4e00\u4e2a\u5206\u5e03\u76f8\u5bf9\u4e8e\u53e6\u4e00\u4e2a\u5206\u5e03\u7684\u504f\u5dee\u3002<\/p>\n \u76f8\u5bf9\u71b5\uff08KL \u6563\u5ea6\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5dee\u5f02\uff0c\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u5de5\u5177\u3002\u5b83\u5177\u6709\u975e\u8d1f\u6027\u548c\u975e\u5bf9\u79f0\u6027\uff0c\u80fd\u591f\u91cf\u5316\u4f7f\u7528\u4e00\u4e2a\u5206\u5e03\u6765\u8fd1\u4f3c\u53e6\u4e00\u4e2a\u5206\u5e03\u65f6\u7684\u4fe1\u606f\u635f\u5931\u3002KL \u6563\u5ea6\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u5e94\u7528\u5e7f\u6cdb\uff0c\u7279\u522b\u662f\u5728\u53d8\u5206\u81ea\u52a8\u7f16\u7801\u5668\u548c\u5f3a\u5316\u5b66\u4e60\u4e2d\uff0c\u8d77\u5230\u4e86\u5173\u952e\u7684\u4f5c\u7528\u3002\u901a\u8fc7\u7406\u89e3\u548c\u5e94\u7528KL \u6563\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u4f18\u5316\u6a21\u578b\uff0c\u63d0\u9ad8\u7b97\u6cd5\u7684\u6027\u80fd\u3002<\/p>\n JS\u6563\u5ea6<\/strong><\/p>\n Jensen-Shannon \u6563\u5ea6\uff08Jensen-Shannon Divergence, JSD\uff09\u662f\u5bf9\u79f0\u7684\u6563\u5ea6\u5ea6\u91cf\uff0c\u57fa\u4e8e KL \u6563\u5ea6\u3002\u5b83\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u7684\u76f8\u4f3c\u6027\uff0c\u5b9a\u4e49\u4e3a\uff1a<\/p>\n $$ $\\mathrm{JSD}$ \u7684\u6027\u8d28:<\/p>\n \u603b\u53d8\u5dee\u6563\u5ea6<\/strong><\/p>\n \u603b\u53d8\u5dee\u6563\u5ea6\uff08Total Variation Divergence\uff09\u662f\u53e6\u4e00\u4e2a\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u5dee\u5f02\u7684\u5ea6\u91cf\u3002\u5b9a\u4e49\u4e3a\uff1a \u603b\u53d8\u5dee\u6563\u5ea6\u7684\u6027\u8d28:<\/p>\n Hellinger\u6563\u5ea6<\/strong><\/p>\n Hellinger \u8ddd\u79bb\u662f\u4e00\u79cd\u57fa\u4e8e\u51e0\u4f55\u89c6\u89d2\u7684\u6563\u5ea6\u5ea6\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u5b9a\u4e49\u4e3a: Hellinger \u8ddd\u79bb\u7684\u6027\u8d28:<\/p>\n \u4ea4\u53c9\u71b5\uff08Cross-Entropy\uff09\u662f\u4fe1\u606f\u8bba\u4e2d\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03 \ud835\udc43 \u548c \ud835\udc44 \u4e4b\u95f4\u76f8\u4f3c\u6027\u7684\u5ea6\u91cf\u3002\u5b83\u901a\u8fc7\u8ba1\u7b97\u4f7f\u7528\u5206\u5e03 \ud835\udc44 \u6765\u8868\u793a\u5206\u5e03 \ud835\udc43 \u65f6\u6240\u9700\u8981\u7684\u989d\u5916\u4fe1\u606f\u91cf\u6765\u8bc4\u4f30\u4e24\u4e2a\u5206\u5e03\u7684\u5dee\u5f02\u3002\u4ea4\u53c9\u71b5\u5e7f\u6cdb\u5e94\u7528\u4e8e\u673a\u5668\u5b66\u4e60\uff0c\u7279\u522b\u662f\u5728\u5206\u7c7b\u95ee\u9898\u4e2d\u7528\u4e8e\u8861\u91cf\u6a21\u578b\u9884\u6d4b\u7684\u51c6\u786e\u6027\u3002<\/p>\n \u5bf9\u4e8e\u79bb\u6563\u6982\u7387\u5206\u5e03 $P$ \u548c $Q$ \uff0c\u4ea4\u53c9\u71b5 $H(P, Q)$ \u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n \u4ea4\u53c9\u71b5\u8861\u91cf\u7684\u662f\u5728\u4f7f\u7528\u4f30\u8ba1\u5206\u5e03 $Q$ \u8868\u793a\u771f\u5b9e\u5206\u5e03 $P$ \u65f6\u6240\u9700\u8981\u7684\u5e73\u5747\u989d\u5916\u4fe1\u606f\u91cf\u3002\u5982\u679c $P$ \u548c $Q$ \u8d8a\u76f8\u4f3c\uff0c\u4ea4\u53c9\u71b5\u503c\u5c31\u8d8a\u5c0f; \u5982\u679c $P$ \u548c $Q$ \u5dee\u5f02\u8d8a\u5927\uff0c\u4ea4\u53c9\u71b5\u503c\u5c31\u8d8a\u5927\u3002<\/p>\n \u4ea4\u53c9\u71b5\u7684\u6027\u8d28<\/strong><\/p>\n \u4ea4\u53c9\u71b5 $H(P, Q)$ \u53ef\u4ee5\u5206\u89e3\u4e3a\u71b5 $H(P)$ \u548c KL \u6563\u5ea6 $D{K L}(P | Q)$ \u4e4b\u548c: \u8fd9\u79cd\u5173\u7cfb\u63ed\u793a\u4e86\u4ea4\u53c9\u6458\u7684\u4e24\u4e2a\u7ec4\u6210\u90e8\u5206: \u4e00\u4e2a\u662f\u5173\u4e8e\u5206\u5e03 $P$ \u7684\u56fa\u6709\u4e0d\u786e\u5b9a\u6027\uff0c\u53e6\u4e00\u4e2a\u662f\u56e0\u4e3a\u5206\u5e03 $Q$\u4e0e $P$ \u4e4b\u95f4\u7684\u4e0d\u5339\u914d\u6240\u5f15\u5165\u7684\u989d\u5916\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n \u5f53\u4e24\u4e2a\u5206\u5e03 $P$ \u548c $Q$ \u5b8c\u5168\u76f8\u540c\u65f6\uff0c\u4ea4\u53c9\u5ae1 $H(P, Q)$ \u7b49\u4e8e\u6458 $H(P)$ \uff0c\u6b64\u65f6 KL\u6563\u5ea6\u4e3a\u96f6: \u8fd9\u8868\u660e\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4f7f\u7528 $Q$ \u8868\u793a $P$ \u6ca1\u6709\u989d\u5916\u7684\u4fe1\u606f\u635f\u5931\u3002<\/p>\n \u5bf9\u4e8e\u591a\u4e2a\u72ec\u7acb\u4e8b\u4ef6\u7684\u8054\u5408\u5206\u5e03\uff0c\u4ea4\u53c9\u6458\u5177\u6709\u53ef\u52a0\u6027\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u4e24\u4e2a\u72ec\u7acb\u4e8b\u4ef6 $\\left(X_1, X_2\\right)$ \u548c $\\left(Y_1, Y_2\\right)$ \uff0c\u5176\u4ea4\u53c9\u6458\u4e3a: \u8fd9\u8868\u660e\u591a\u4e2a\u72ec\u7acb\u4e8b\u4ef6\u7684\u8054\u5408\u5206\u5e03\u7684\u4ea4\u53c9\u6458\u7b49\u4e8e\u5404\u4e2a\u72ec\u7acb\u4e8b\u4ef6\u4ea4\u53c9\u6458\u7684\u603b\u548c\u3002<\/p>\n \u4ea4\u53c9\u71b5 $H(P, Q)$ \u662f\u8861\u91cf\u4e24\u4e2a\u6982\u7387\u5206\u5e03 $P$ \u548c $Q$ \u4e4b\u95f4\u76f8\u4f3c\u6027\u7684\u5ea6\u91cf\u3002\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u88ab\u5e7f\u6cdb\u7528\u4e8e\u5206\u7c7b\u95ee\u9898\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u51e0\u4e2a\u539f\u56e0\uff1a<\/p>\n \u6982\u7387\u89e3\u91ca\uff1a\u4ea4\u53c9\u71b5\u53ef\u4ee5\u5f88\u597d\u5730\u5904\u7406\u6982\u7387\u8f93\u51fa\u3002\u5728\u5206\u7c7b\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u5e0c\u671b\u8f93\u51fa\u4e00\u4e2a\u6982\u7387\u5206\u5e03\uff0c\u800c\u4ea4\u53c9\u71b5\u635f\u5931\u76f4\u63a5\u8861\u91cf\u4e86\u9884\u6d4b\u6982\u7387\u5206\u5e03\u4e0e\u771f\u5b9e\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u5dee\u5f02\u3002<\/p>\n<\/li>\n \u5bf9\u6570\u635f\u5931\uff1a\u4ea4\u53c9\u71b5\u5305\u62ec\u5bf9\u6570\u51fd\u6570\uff0c\u8fd9\u4f1a\u4f7f\u5f97\u5f53\u6a21\u578b\u9884\u6d4b\u63a5\u8fd1\u4e8e\u771f\u5b9e\u6807\u7b7e\u65f6\uff0c\u635f\u5931\u8f83\u4f4e\uff1b\u800c\u5f53\u9884\u6d4b\u9519\u8bef\u8f83\u5927\u65f6\uff0c\u635f\u5931\u4f1a\u6025\u5267\u589e\u5927\u3002\u8fd9\u79cd\u7279\u6027\u80fd\u591f\u6709\u6548\u5730\u60e9\u7f5a\u9519\u8bef\u9884\u6d4b\uff0c\u5e76\u6709\u52a9\u4e8e\u6a21\u578b\u5feb\u901f\u6536\u655b\u3002<\/p>\n<\/li>\n<\/ul>\n \u4e92\u4fe1\u606f\u7684\u5b9a\u4e49<\/strong><\/p>\n \u4e92\u4fe1\u606f (Mutual Information, MI) \u662f\u4fe1\u606f\u8bba\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u4e92\u4f9d\u8d56\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4e92\u4fe1\u606f\u8861\u91cf\u7684\u662f\u77e5\u9053\u4e00\u4e2a\u53d8\u91cf\u7684\u4fe1\u606f\u91cf\u5bf9\u51cf\u5c11\u53e6\u4e00\u4e2a\u53d8\u91cf\u7684\u4e0d\u786e\u5b9a\u6027\u6240\u505a\u7684\u8d21\u732e\u3002\u5bf9\u4e8e\u4e24\u4e2a\u79bb\u6563\u968f\u673a\u53d8\u91cf $X$ \u548c $Y$ \uff0c\u5176\u4e92\u4fe1\u606f $I(X ; Y)$ \u5b9a\u4e49\u4e3a: \u5176\u4e2d:<\/p>\n \u4e92\u4fe1\u606f\u7684\u6027\u8d28<\/strong><\/p>\n \u4e92\u4fe1\u606f\u7684\u5e94\u7528<\/strong><\/p>\n \u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u4e92\u4fe1\u606f\uff08Mutual Information, MI\uff09\u6700\u5e38\u89c1\u7684\u5e94\u7528\u662f\u53ef\u4ee5\u7528\u4e8e\u7279\u5f81\u9009\u62e9\uff0c\u5e2e\u52a9\u9009\u62e9\u90a3\u4e9b\u4e0e\u76ee\u6807\u53d8\u91cf\u5177\u6709\u8f83\u5927\u76f8\u5173\u6027\u7684\u7279\u5f81\u3002\u901a\u8fc7\u6700\u5927\u5316\u7279\u5f81\u548c\u76ee\u6807\u53d8\u91cf\u4e4b\u95f4\u7684\u4e92\u4fe1\u606f\uff0c\u6211\u4eec\u53ef\u4ee5\u7b5b\u9009\u51fa\u5bf9\u9884\u6d4b\u6700\u6709\u7528\u7684\u7279\u5f81\uff0c\u4ece\u800c\u63d0\u9ad8\u6a21\u578b\u7684\u6027\u80fd\u3002 \u5047\u8bbe\u6211\u4eec\u6709\u8311\u5c3e\u82b1\u6570\u636e\u96c6\uff0c\u5176\u4e2d\u76ee\u6807\u53d8\u91cf $Y$ \u8868\u793a\u603b\u5c3e\u82b1\u7684\u7c7b\u522b\uff0c\u7279\u5f81\u53d8\u91cf $X_1, X_2, X_3, X_4$ \u5206\u522b\u8868 $I\\left(X_i ; Y\\right)$ \uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u4e92\u4fe1\u606f\u503c\u8f83\u5927\u7684\u7279\u5f81\u3002\u4e0b\u9762\u662f\u5982\u4f55\u5728\u603b\u5c3e\u82b1\u6570\u636e\u96c6\u4e2d\u5e94\u7528\u8fd9\u4e00\u8fc7\u7a0b\u7684\u5177\u4f53\u793a\u4f8b:<\/p>\n \n \u4ece\u4e0a\u8868\u4e2d\u53ef\u4ee5\u770b\u5230\uff0c\u201c\u82b1\u74e3\u957f\u5ea6\u201d\u548c\u201c\u82b1\u74e3\u5bbd\u5ea6\u201d\u4e0e\u76ee\u6807\u53d8\u91cf\uff08\u9e22\u5c3e\u82b1\u7c7b\u522b\uff09\u7684\u4e92\u4fe1\u606f\u503c\u8f83\u5927\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u4eec\u5bf9\u5206\u7c7b\u6700\u6709\u7528\u3002\u800c\u201c\u82b1\u843c\u957f\u5ea6\u201d\u548c\u201c\u82b1\u843c\u5bbd\u5ea6\u201d\u7684\u4e92\u4fe1\u606f\u503c\u8f83\u5c0f\uff0c\u5bf9\u5206\u7c7b\u7684\u8d21\u732e\u8f83\u5c0f\u3002<\/p>\n \u56e0\u6b64\uff0c\u901a\u8fc7\u9009\u62e9\u4e92\u4fe1\u606f\u503c\u8f83\u5927\u7684\u7279\u5f81\uff0c\u6211\u4eec\u53ef\u4ee5\u7b5b\u9009\u51fa\u5bf9\u9884\u6d4b\u6700\u6709\u7528\u7684\u7279\u5f81\uff0c\u4ece\u800c\u63d0\u5347\u6a21\u578b\u7684\u6027\u80fd\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u4ee5\u4ec5\u4f7f\u7528\u201c\u82b1\u74e3\u957f\u5ea6\u201d\u548c\u201c\u82b1\u74e3\u5bbd\u5ea6\u201d\u6765\u8bad\u7ec3\u5206\u7c7b\u6a21\u578b\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u7684\u51c6\u786e\u6027\u548c\u6548\u7387\u3002<\/p>\n \u8fd9\u4e2a\u8fc7\u7a0b\u5c55\u793a\u4e86\u5982\u4f55\u5229\u7528\u4e92\u4fe1\u606f\u8fdb\u884c\u7279\u5f81\u9009\u62e9\uff0c\u4ece\u800c\u5728\u5206\u7c7b\u95ee\u9898\u4e2d\u63d0\u9ad8\u6a21\u578b\u6027\u80fd\u3002<\/p>\n \u8fd9\u4e9b\u4e0d\u7b49\u5f0f\u5728\u4fe1\u606f\u8bba\u4e2d\u8d77\u7740\u5173\u952e\u4f5c\u7528\uff0c\u63d0\u4f9b\u4e86\u5404\u79cd\u4fe1\u606f\u91cf\u5ea6\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u548c\u754c\u9650\u3002\u901a\u8fc7\u8fd9\u4e9b\u4e0d\u7b49\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u8bb8\u591a\u91cd\u8981\u7684\u7406\u8bba\u7ed3\u679c\uff0c\u5e76\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u7528\u4e8e\u8bc4\u4f30\u548c\u4f18\u5316\u4fe1\u606f\u5904\u7406\u548c\u4f20\u8f93\u7cfb\u7edf\u3002\u8fd9\u4e9b\u4e0d\u7b49\u5f0f\u4e0d\u4ec5\u5728\u4fe1\u606f\u8bba\u4e2d\u5177\u6709\u7406\u8bba\u610f\u4e49\uff0c\u800c\u4e14\u5728\u673a\u5668\u5b66\u4e60\u3001\u7edf\u8ba1\u5b66\u548c\u6570\u636e\u79d1\u5b66\u7b49\u9886\u57df\u4e5f\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u4ee5\u4e0b\u662f\u51e0\u79cd\u5e38\u89c1\u7684\u6570\u636e\u56fe\u5f62\u5316\u8868\u793a\u65b9\u5f0f\uff1a<\/p>\n Jensen \u4e0d\u7b49\u5f0f<\/p>\n<\/li>\n Chebyshev \u4e0d\u7b49\u5f0f<\/p>\n<\/li>\n Pinsker \u4e0d\u7b49\u5f0f<\/p>\n<\/li>\n<\/ul>\n Jensen \u4e0d\u7b49\u5f0f\u662f\u51f8\u51fd\u6570\u4e0e\u671f\u671b\u503c\u4e4b\u95f4\u7684\u91cd\u8981\u5173\u7cfb\uff0c\u5728\u4fe1\u606f\u8bba\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5982\u679c $\\phi$\u662f\u4e00\u4e2a\u51f8\u51fd\u6570\uff0c\u5e76\u4e14 $X$ \u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48 Jensen \u4e0d\u7b49\u5f0f\u8868\u660e: Jensen \u4e0d\u7b49\u5f0f\u5728\u4fe1\u606f\u8bba\u548c\u6982\u7387\u8bba\u4e2d\u6709\u591a\u79cd\u5e94\u7528\uff0c\u5176\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u5e94\u7528\u4e3a\u53d8\u5206\u63a8\u65ad<\/strong>\u3002<\/p>\n \u5728\u53d8\u5206\u63a8\u65ad\uff08Variational Inference, VI\uff09\u4e2d\uff0cJensen \u4e0d\u7b49\u5f0f\u7528\u4e8e\u63a8\u5bfc\u8bc1\u636e\u4e0b\u754c\uff08Evidence Lower Bound, ELBO\uff09\uff0c\u4ece\u800c\u4f7f\u6211\u4eec\u80fd\u591f\u4ee5\u4f18\u5316\u7684\u65b9\u5f0f\u8fdb\u884c\u8fd1\u4f3c\u63a8\u65ad\u3002<\/p>\n \u53d8\u5206\u63a8\u65ad\u7684\u76ee\u6807\u662f\u8fd1\u4f3c\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03 $P(Z \\mid X)$ \uff0c\u5176\u4e2d $X$ \u662f\u89c2\u6d4b\u6570\u636e\uff0c $Z$ \u662f\u6f5c\u5728\u53d8\u91cf\u3002\u76f4\u63a5\u8ba1\u7b97\u540e\u9a8c\u5206\u5e03\u901a\u5e38\u662f\u4e0d\u53ef\u884c\u7684\uff0c\u56e0\u4e3a\u5b83\u6d89\u53ca\u590d\u6742\u7684\u79ef\u5206\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8fd9\u4e2a\u79ef\u5206\u7684\u96be\u70b9\u5728\u4e8e:<\/p>\n \u56e0\u6b64\uff0c\u6211\u4eec\u901a\u8fc7\u5f15\u5165\u4e00\u4e2a\u53d8\u5206\u5206\u5e03 $q(Z)$ \u6765\u8fd1\u4f3c $p(Z \\mid X)$ \uff0c\u53ef\u4ee5\u5f97\u5230: \u5e94\u7528 Jensen \u4e0d\u7b49\u5f0f\uff0c\u6211\u4eec\u5f97\u5230: \u56e0\u4e3a\u5bf9\u6570\u662f\u51f9\u51fd\u6570\uff0cJensen\u4e0d\u7b49\u5f0f\u7ed9\u51fa\u4e86\u8fd9\u4e2a\u4e0b\u754c\u3002\u73b0\u5728\u6211\u4eec\u5c06\u53f3\u8fb9\u7684\u8868\u8fbe\u5f0f\u5206\u89e3: \u6211\u4eec\u5c06\u79ef\u5206\u5206\u89e3\u4e3a\u4e24\u4e2a\u90e8\u5206: \u7b2c\u4e00\u4e2a\u90e8\u5206\u662f\u91cd\u6784\u8bef\u5dee\u7684\u671f\u671b\uff1a \u7b2c\u4e8c\u4e2a\u90e8\u5206\u662fKL\u6563\u5ea6\uff1a \u56e0\u6b64\uff0c\u6211\u4eec\u5f97\u5230\u8bc1\u636e\u4e0b\u754c (ELBO) \u7684\u6700\u7ec8\u8868\u8fbe\u5f0f: \u8bc1\u636e\u4e0b\u754c (ELBO) \u8868\u793a\u4e3a: \u8fd9\u4e2aELBO\u7684\u4f18\u5316\u76ee\u6807\u5c31\u662f\u5728\u7ed9\u5b9a\u6570\u636e $X$ \u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u5230\u6700\u4f18\u7684\u53d8\u5206\u5206\u5e03 $q(Z)$ \uff0c\u4f7f\u5f97\u8be5\u4e0b\u754c\u6700\u5927\u5316\u3002\u6700\u5927\u5316\u8fd9\u4e2a\u4e0b\u754c\u76f8\u5f53\u4e8e\u6700\u5c0f\u5316KL\u6563\u5ea6\uff0c\u4f7f\u5f97 $q(Z)$ \u66f4\u63a5\u8fd1\u771f\u5b9e\u540e\u9a8c\u5206\u5e03 $p(Z \\mid X)$ \u3002\u8fd9\u5c06\u5728\u53d8\u5206\u81ea\u7f16\u7801\u5668VAE\u4e2d\u8fdb\u884c\u8be6\u7ec6\u8bb2\u89e3\u3002<\/p>\n Jensen\u4e0d\u7b49\u5f0f\u53e6\u4e00\u4e2a\u91cd\u8981\u5e94\u7528\u4e3a\u8bc1\u660eKL\u6563\u5ea6\u4e3a\u5927\u4e8e0\u7684\u6b63\u6570<\/strong>\uff1a<\/p>\n \u5177\u4f53\u6765\u8bf4\uff0cKL\u6563\u5ea6\u7684\u5b9a\u4e49\u662f: \u7531\u4e8e\u5bf9\u6570\u51fd\u6570\u662f\u4e00\u4e2a\u51f8\u51fd\u6570\uff0c\u6240\u4ee5\u6839\u636eJensen\u4e0d\u7b49\u5f0f\uff0c\u53ef\u4ee5\u5f97\u5230: Chebyshev \u4e0d\u7b49\u5f0f<\/strong> <\/p>\n Chebyshev\u7528\u4e8e\u91cf\u5316\u968f\u673a\u53d8\u91cf\u504f\u79bb\u5176\u5747\u503c\u7684\u6982\u7387\u3002\u8be5\u4e0d\u7b49\u5f0f\u6307\u51fa\uff0c\u5bf9\u4e8e\u4efb\u4f55\u968f\u673a\u53d8\u91cf\uff0c\u65e0\u8bba\u5176\u5206\u5e03\u5982\u4f55\uff0c\u53ea\u8981\u77e5\u9053\u5b83\u7684\u5747\u503c\u548c\u65b9\u5dee\uff0c\u5c31\u53ef\u4ee5\u754c\u5b9a\u5b83\u504f\u79bb\u5747\u503c\u7684\u6982\u7387\u4e0a\u754c\u3002<\/p>\n \u516c\u5f0f\u5b9a\u4e49<\/p>\n Chebyshev \u4e0d\u7b49\u5f0f\u8868\u660e\uff0c\u5bf9\u4e8e\u4efb\u4f55\u968f\u673a\u53d8\u91cf $X$ \u4ee5\u53ca\u4efb\u4f55\u6b63\u6570 $k>0$ \uff0c\u90fd\u6709: \u5176\u4e2d:<\/p>\n Chebyshev \u4e0d\u7b49\u5f0f\u7684\u76f4\u89c2\u89e3\u91ca\u662f\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e00\u4e2a\u4fdd\u5b88\u7684\u4f30\u8ba1\uff0c\u8bf4\u660e\u968f\u673a\u53d8\u91cf $X$ \u504f\u79bb\u5176\u5747\u503c $\\mu$ \u8d85\u8fc7 $k \\sigma$ \u7684\u6982\u7387\u4e0d\u4f1a\u8d85\u8fc7 $\\frac{1}{k^2}$ \u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u968f\u673a\u53d8\u91cf $X$ \u6709\u81f3\u5c11 $1-\\frac{1}{k^2}$ \u7684\u6982\u7387\u5728\u5747\u503c $\\mu$ \u7684 $k \\sigma$ \u8303\u56f4\u5185\u3002<\/p>\n \u4e3e\u4e2a\u4f8b\u5b50\uff1a\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u968f\u673a\u53d8\u91cf $X$ \uff0c\u5176\u671f\u671b $\\mu=50$ \uff0c\u6807\u51c6\u5dee $\\sigma=5$ \u3002\u6211\u4eec\u5e0c\u671b\u4f30\u8ba1 $X$ \u504f\u79bb\u5176\u5747\u503c\u8d85\u8fc7 10 \u7684\u6982\u7387\uff0c\u5373 $k=2$ \u65f6\u7684\u60c5\u51b5\u3002<\/p>\n \u6839\u636e Chebyshev \u4e0d\u7b49\u5f0f: \u8fd9\u610f\u5473\u7740\u968f\u673a\u53d8\u91cf $X$ \u504f\u79bb\u5176\u5747\u503c 50 \u8d85\u8fc7 10 \u7684\u6982\u7387\u4e0d\u8d85\u8fc7 0.25 \uff08\u5373 $25 \\%$ )\u3002<\/p>\n Pinsker \u4e0d\u7b49\u5f0f<\/strong><\/p>\n Pinsker \u4e0d\u7b49\u5f0f\u5728\u4fe1\u606f\u8bba\u548c\u6982\u7387\u8bba\u4e2d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u5de5\u5177\uff0c\u7528\u4e8e\u91cf\u5316\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u603b\u53d8\u5206\u6563\u5ea6\uff08Total Variation Distance, TVD\uff09\u548c\u5b83\u4eec\u7684\u76f8\u5bf9\u71b5\uff08\u5373 Kullback-Leibler \u6563\u5ea6\uff0cKL \u6563\u5ea6\uff09\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n \u516c\u5f0f\u8868\u793a\uff1a<\/p>\n \u5177\u4f53\u6765\u8bf4\uff0cPinsker \u4e0d\u7b49\u5f0f\u8868\u660e\uff0c\u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u6982\u7387\u5206\u5e03 $P$ \u548c $Q$ \uff0c\u5b83\u4eec\u7684\u603b\u53d8\u5dee\u8ddd\u79bb $\\delta(P, Q)$ \u548c $\\mathrm{KL}$ \u6563\u5ea6$D_{\\mathrm{KL}}(P | Q)$ \u6ee1\u8db3\u4ee5\u4e0b\u5173\u7cfb\uff1a \u5176\u4e2d\uff0c\u603b\u53d8\u5dee\u8ddd\u79bb\u5b9a\u4e49\u4e3a: KL \u6563\u5ea6\u5b9a\u4e49\u4e3a: \u76f4\u89c2\u89e3\u91ca Pinsker \u4e0d\u7b49\u5f0f\u610f\u5473\u7740\uff0c\u5c3d\u7ba1 KL \u6563\u5ea6\u548c\u603b\u53d8\u5dee\u8ddd\u79bb\u5ea6\u91cf\u7684\u89d2\u5ea6\u4e0d\u540c\uff0c\u4f46\u5b83\u4eec\u5728\u8861\u91cf\u6982\u7387\u5206\u5e03\u5dee\u5f02\u65f6\u662f\u76f8\u5173\u7684\u3002\u8fd9\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4ece\u4fe1\u606f\u8bba\u89d2\u5ea6\uff08\u901a\u8fc7 KL \u6563\u5ea6\uff09\u4f30\u8ba1\u6982\u7387\u8bba\u89d2\u5ea6\uff08\u901a\u8fc7\u603b\u53d8\u5206\u6563\u5ea6\uff09\u5dee\u5f02\u7684\u5de5\u5177\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002<\/p>\n \u7edf\u8ba1\u5b66\u65e8\u5728\u6839\u636e\u6570\u636e\u6837\u672c\u63a8\u6d4b\u603b\u60c5\u51b5\u3002\u5927\u90e8\u5206\u7edf\u8ba1\u5206\u6790\u90fd\u57fa\u4e8e\u6982\u7387\uff0c\u6240\u4ee5\u8fd9\u4e24\u65b9\u9762\u7684\u5185\u5bb9\u901a\u5e38\u517c\u800c\u6709\u4e4b\u3002<\/p>\n Deep Learning Math \u4fe1\u606f\u8bba\uff08Information Theory\uff09 \u4fe1\u606f\u8bba\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u81f3\u5173\u91cd […]<\/p>\n","protected":false},"author":1,"featured_media":1644,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1516","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14"],"_links":{"self":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1516","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1516"}],"version-history":[{"count":23,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1516\/revisions"}],"predecessor-version":[{"id":1691,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1516\/revisions\/1691"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/media\/1644"}],"wp:attachment":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1516"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1516"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}} \u4fe1\u606f\u91cf\u4e0e\u71b5\uff08Information and Entropy\uff09<\/h2>\n
\n
\n \u4fe1\u606f\u91cf\u7684\u5b9a\u4e49<\/h3>\n
\n$$
\nI(x)=-\\log _b P(x)
\n$$<\/p>\n\n
\n$$
\nI(x)=-\\log _2(0.5)=-(-1)=1 \\text { bit }
\n$$<\/p>\n
\n$$
\nI(x)=-\\log _e(0.5) \\approx 0.693 \\text { nats }
\n$$<\/p>\n \u71b5\u7684\u5b9a\u4e49<\/h3>\n
\n
\n$$
\nH(X)=-\\sum_{x \\in X} P(x) \\log _b P(x)
\n$$<\/p>\n\n
\n$$
\nH(X)=\\mathbb{E}[I(X)]=\\sum_{x \\in X} P(x) I(x)
\n$$<\/p>\n\n
\n
\n
\n
\n$$
\nH(X)=-\\left[1 \\log _b 1\\right]=0
\n$$<\/p>\n\n
\n
\n$$
\nH(X)=-\\sum_{i=1}^n \\frac{1}{n} \\log _b \\frac{1}{n}=\\log _b n
\n$$<\/p>\n\u8054\u5408\u71b5<\/h3>\n
\n$$
\nH(X, Y)=-\\sum_{x \\in X} \\sum_{y \\in Y} P(x, y) \\log _b P(x, y)
\n$$<\/p>\n\n
\n$$
\nH(X, Y)=H(X)+H(Y)
\n$$<\/p>\n
\n\u5982\u679c $X$ \u548c $Y$ \u662f\u76f8\u5173\u7684\u968f\u673a\u53d8\u91cf\uff0c\u5b83\u4eec\u7684\u8054\u5408\u6982\u7387\u5206\u5e03 $P(x, y)$ \u4e0d\u80fd\u7b80\u5355\u5730\u5206\u89e3\u4e3a\u5355\u53d8\u91cf\u6982\u7387\u7684\u4e58\u79ef\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8054\u5408\u71b5 $H(X, Y)$ \u4e0d\u7b49\u4e8e\u5355\u53d8\u91cf\u71b5\u7684\u548c\uff0c\u800c\u662f\u8003\u8651\u4e86\u4e24\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\u3002<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729193930787.png\")
\n<\/p>\n\n
\n
\n$$
\nP(0,0) \\log _2 P(0,0)=0.1 \\log _2 0.1 \\approx 0.1 \\cdot(-3.32)=-0.332
\n$$<\/li>\n
\n$$
\nP(0,1) \\log _2 P(0,1)=0.4 \\log _2 0.4 \\approx 0.4 \\cdot(-1.32)=-0.528
\n$$<\/li>\n
\n$$
\nP(1,0) \\log _2 P(1,0)=0.2 \\log _2 0.2 \\approx 0.2 \\cdot(-2.32)=-0.464
\n$$<\/li>\n
\n$$
\nP(1,1) \\log _2 P(1,1)=0.3 \\log _2 0.3 \\approx 0.3 \\cdot(-1.737)=-0.5211
\n$$<\/li>\n<\/ul>\n<\/li>\n
\n$$
\nH(X, Y)=-(-0.332-0.528-0.464-0.5211) \\approx 1.8451 \\text { bits }
\n$$<\/p>\n<\/li>\n<\/ol>\n\u6761\u4ef6\u71b5<\/h3>\n
\n$$
\nH(Y \\mid X)=-\\sum_{x \\in X} \\sum_{y \\in Y} P(x, y) \\log _b P(y \\mid x)
\n$$<\/p>\n\n
\uff1a<\/p>\n
\n
\n$$
\nH(X, Y)=H(X)+H(Y \\mid X)
\n$$<\/p>\n\n
\u76f8\u5bf9\u71b5\uff08Kullback-Leibler \u6563\u5ea6\uff0c\u7b80\u79f0 KL \u6563\u5ea6\uff09<\/h3>\n
\n$$
\nD_{K L}(P | Q)=\\sum_{x \\in X} P(x) \\log _b \\frac{P(x)}{Q(x)}
\n$$<\/p>\n\n
\n
\n$$
\nD_{K L}(P | Q) \\geq 0
\n$$<\/p>\n\n
\n$$
\nD_{K L}(P | Q) \\neq D_{K L}(Q | P)
\n$$<\/p>\n
\nJ S D(P | Q)=\\frac{1}{2} D_{K L}(P | M)+\\frac{1}{2} D_{K L}(Q | M)
\n$$
\n\u5176\u4e2d $M=\\frac{1}{2}(P+Q)$.
\nJSD \u7684\u6027\u8d28:<\/p>\n\n
\n$$
\nD_{T V}(P, Q)=\\frac{1}{2} \\sum_{x \\in X}|P(x)-Q(x)|
\n$$<\/p>\n\n
\n$$
\nH(P, Q)=\\frac{1}{\\sqrt{2}}\\left(\\sum_{x \\in X}(\\sqrt{P(x)}-\\sqrt{Q(x)})^2\\right)^{\\frac{1}{2}}
\n$$<\/p>\n\n
\u4ea4\u53c9\u71b5<\/h3>\n
\n$$
\nH(P, Q)=-\\sum_{x \\in X} P(x) \\log _b Q(x)
\n$$<\/p>\n\n
\n
\n$$
\nH(P, Q)=H(P)+D\\<\/em>{K L}(P | Q)
\n$$
\n\u5176\u4e2d:<\/p>\n\n
\n
\n$$
\nH(P, Q)=H(P)
\n$$<\/p>\n\n
\n$$
\nH\\left(P_1, P_2 ; Q_1, Q_2\\right)=H\\left(P_1, Q_1\\right)+H\\left(P_2, Q_2\\right)
\n$$<\/p>\n\n
![](\"https:\/\/img.icons8.com\/plasticine\/100\/000000\/mind-map.png\")
\u4e92\u4fe1\u606f<\/h2>\n
\n
\n$$
\nI(X ; Y)=\\sum_{x \\in X} \\sum_{y \\in Y} P(x, y) \\log \\frac{P(x, y)}{P(x) P(y)}
\n$$<\/p>\n\n
\n
\n
\n$$
\nI(X ; Y)=H(X)+H(Y)-H(X, Y)
\n$$<\/li>\n
\n$$
\nI(X ; Y)=H(X)-H(X \\mid Y)=H(Y)-H(Y \\mid X)
\n$$<\/li>\n<\/ul>\n
\n\u4e3e\u4e2a\uff1a<\/p>\n
from sklearn.datasets import load_iris\nimport pandas as pd\n\n# \u52a0\u8f7d\u9e22\u5c3e\u82b1\u6570\u636e\u96c6\niris = load_iris()\nX = pd.DataFrame(iris.data, columns=iris.feature_names)\ny = pd.Series(iris.target, name='species')\n\nfrom sklearn.feature_selection import mutual_info_classif\n\n# \u8ba1\u7b97\u6bcf\u4e2a\u7279\u5f81\u4e0e\u76ee\u6807\u53d8\u91cf\u4e4b\u95f4\u7684\u4e92\u4fe1\u606f\nmi = mutual_info_classif(X, y)\n\n# \u5c06\u4e92\u4fe1\u606f\u7ed3\u679c\u4e0e\u7279\u5f81\u540d\u79f0\u5bf9\u5e94\nmi_df = pd.DataFrame({'Feature': X.columns, 'Mutual Information': mi})\n\n# \u6839\u636e\u4e92\u4fe1\u606f\u503c\u6392\u5e8f\nmi_df = mi_df.sort_values(by='Mutual Information', ascending=False)\n\n# \u663e\u793a\u7ed3\u679c\nprint(mi_df)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729194012809.png\")
\n<\/p>\n \u4e0d\u7b49\u5f0f<\/h2>\n
\n\n
Jensen \u4e0d\u7b49\u5f0f<\/h3>\n
\n$$
\n\\phi(\\mathbb{E}[X]) \\leq \\mathbb{E}[\\phi(X)]
\n$$<\/p>\n\n
\n$$
\n\\log p(X)=\\log \\int \\frac{p(X, Z)}{q(Z)} q(Z) d Z
\n$$<\/p>\n
\n$$
\n\\log p(X) \\geq \\int q(Z) \\log \\frac{p(X, Z)}{q(Z)} d Z
\n$$<\/p>\n
\n$$
\n\\log p(X) \\geq \\int q(Z) \\log \\frac{p(X \\mid Z) p(Z)}{q(Z)} d Z
\n$$<\/p>\n
\n$$
\n\\log p(X) \\geq \\int q(Z) \\log p(X \\mid Z) d Z+\\int q(Z) \\log \\frac{p(Z)}{q(Z)} d Z
\n$$<\/p>\n
\n$$
\n\\mathbb{E}_{q(Z)}[\\log p(X \\mid Z)]
\n$$<\/p>\n
\n$$
\n-\\mathrm{KL}(q(Z) | p(Z))
\n$$<\/p>\n
\n$$
\n\\log p(X) \\geq \\mathbb{E}_{q(Z)}[\\log p(X \\mid Z)]-\\operatorname{KL}(q(Z) | p(Z))
\n$$<\/p>\n
\n$$
\n\\mathcal{L}=\\mathbb{E}_{q(Z)}[\\log p(X \\mid Z)]-\\operatorname{KL}(q(Z) | p(Z))
\n$$<\/p>\n
\n$$
\nK L(q | p)=\\int q(z) \\log \\left(\\frac{q(z)}{p(z)}\\right) d z
\n$$<\/p>\n
\n$$
\n\\int q(z) \\log \\left(\\frac{q(z)}{p(z)}\\right) d z \\geq \\log \\left(\\int q(z) \\frac{q(z)}{p(z)} d z\\right)=\\log (1)=0
\n$$<\/p>\n
\n$$
\nP(|X-\\mu| \\geq k \\sigma) \\leq \\frac{1}{k^2}
\n$$<\/p>\n\n
\n$$
\nP(|X-50| \\geq 10) \\leq \\frac{1}{2^2}=\\frac{1}{4}=0.25
\n$$<\/p>\n
\n$$
\n\\delta(P, Q) \\leq \\sqrt{\\frac{1}{2} D_{\\mathrm{KL}}(P | Q)}
\n$$<\/p>\n
\n$$
\n\\delta(P, Q)=\\frac{1}{2} \\sum_x|P(x)-Q(x)|
\n$$<\/p>\n
\n$$
\nD_{\\mathrm{KL}}(P | Q)=\\sum_x P(x) \\log \\frac{P(x)}{Q(x)}
\n$$<\/p>\n
\nPinsker \u4e0d\u7b49\u5f0f\u63d0\u4f9b\u4e86\u4e00\u79cd\u624b\u6bb5\uff0c\u5c06\u4e24\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u95f4\u7684\u201c\u8ddd\u79bb\u201d\u4ece\u4fe1\u606f\u7406\u8bba\u7684\u5ea6\u91cf\uff08KL \u6563\u5ea6\uff09\u8f6c\u6362\u4e3a\u6982\u7387\u8bba\u7684\u5ea6\u91cf\uff08\u603b\u53d8\u5206\u6563\u5ea6\uff09\u3002\u5177\u4f53\u6765\u8bf4\uff1a<\/p>\n\n
\u7edf\u8ba1\u5b66<\/h2>\n
\n
\n![](\"https:\/\/img.icons8.com\/dusk\/64\/000000\/prize.png\")
Credits<\/h2>\n
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