{"id":30750,"date":"2021-12-08T23:04:30","date_gmt":"2021-12-08T15:04:30","guid":{"rendered":"https:\/\/swarma.org\/?p=30750"},"modified":"2021-12-08T23:04:30","modified_gmt":"2021-12-08T15:04:30","slug":"%e4%bb%80%e4%b9%88%e6%98%af%e6%9d%a1%e4%bb%b6%e4%ba%92%e4%bf%a1%e6%81%af-%e9%9b%86%e6%99%ba%e7%99%be%e7%a7%91","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=30750","title":{"rendered":"\u4ec0\u4e48\u662f\u6761\u4ef6\u4e92\u4fe1\u606f | \u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
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\u672c\u6587\u662f\u5bf9\u96c6\u667a\u767e\u79d1\u4e2d\u201c\u6761\u4ef6\u4e92\u4fe1\u606f<\/span>\u201d\u8bcd\u6761\u7684\u6458\u5f55\uff0c\u53c2\u8003\u8d44\u6599\u53ca\u76f8\u5173\u8bcd\u6761\u8bf7\u53c2\u9605\u767e\u79d1\u8bcd\u6761\u539f\u6587\u3002<\/span><\/section>\n

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\u5341\u3001\u767e\u79d1\u9879\u76ee\u5fd7\u613f\u8005\u62db\u52df<\/strong><\/strong><\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
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\u5bf9\u4e8e\u5177<\/span>\u6709\u652f\u6301\u96c6 Probability theory <\/span><\/h2>\n

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\u5bf9\u4e8e\u5177\u6709\u652f\u6301\u96c6X,Y\u548cZ\u7684\u79bb\u6563\u968f\u673a\u53d8\u91cf<\/span>\u6761\u4ef6\u4e92\u4fe1\u606f<\/span>I(X;Y|Z)\u5982\u4e0b:<\/span>
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\u5bf9\u4e8e\u5177\u6709\u652f\u6301\u96c6X,Y\u548cZ\u7684\uff08\u7edd\u5bf9\uff09\u8fde\u7eed\u968f\u673a\u53d8\u91cf<\/span>\u6761\u4ef6\u4e92\u4fe1\u606fI(X;Y|Z)\u5982\u4e0b:<\/span>
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\u5176\u4e2d\u8fb9\u7f18\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u8054\u5408\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u548c\uff08\u6216\uff09\u6761\u4ef6\u6982\u7387\u5bc6\u5ea6\u51fd\u6570 \u53ef\u4ee5\u7531p\u52a0\u4e0a\u9002\u5f53\u7684\u4e0b\u6807\u8868\u793a\u3002\u8fd9\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a<\/span><\/p>\n


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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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\u540c\u65f6\u6211\u4eec\u4e5f\u53ef\u4ee5\u5c06\u8054\u5408\u548c\u6761\u4ef6\u71b5\u5199\u4e3a<\/span>\uff1aI(X;Y|Z) = H(X,Z) + H(Y,Z) – H(X,Y,Z) – H(Z) = H(X|Z) – H()X|Y,Z) = H(X|Z) + H(Y|Z) – H(X,Y|Z)<\/span>
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\u8fd9\u4e48\u8868\u8fbe\u4ee5\u663e\u793a\u5176\u4e0e\u4e92\u4fe1\u606f\u7684\u5173\u7cfb\uff1aI(X;Y|Z) = I(X;Y,Z) – I(X;Z)<\/span><\/p>\n


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\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8868\u8fbe\u5f0f\u88ab\u91cd\u65b0\u6574\u7406\u4e3a\u201c\u4e92\u4fe1\u606f\u7684\u94fe\u5f0f\u6cd5\u5219\u201d:<\/span><\/p>\n

I(X;Y|Z) = I(X;Z) + I(X;Y|Z)<\/span><\/p>\n


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\u4e0a\u8ff0\u5f0f\u5b50\u7684\u53e6\u4e00\u79cd\u7b49\u4ef7\u5f62\u5f0f\u662f\uff1aI(X;Y|Z) = H(Z|X) + H(X) + H(Z|Y) + H(Y) -H(Z|X,Y) – H(X,Y) – H(Z) = I(X;Y) + H(Z|X) + H(Z|Y) – H(Z|X,Y)<\/span><\/p>\n


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\u6216\u4f5c\u4e3a\u66f4\u7b80\u5355\u7684KL\u6563\u5ea6\u7684\u671f\u671b\u503c\uff1a<\/span><\/p>\n


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\u6761\u4ef6\u4e92\u4fe1\u606f\u7684\u5176\u4ed6\u901a\u7528\u5b9a\u4e49\uff08\u9002\u7528\u4e8e\u5177\u6709\u8fde\u7eed\u6216\u5176\u4ed6\u4efb\u610f\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff09\u5c06\u53d6\u51b3\u4e8e \u6b63<\/span>\u5219\u6761\u4ef6\u6982\u7387 Regular conditional probability \u7684\u6982\u5ff5\u3002<\/span><\/h2>\n

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\u4e3a\u4e00\u4e2a \u6982\u7387\u7a7a\u95f4 Probability space \uff0c\u5e76\u5c06\u968f\u673a\u53d8\u91cfX,Y\u548cZ\u5206\u522b\u5b9a\u4e49\u4e3a\u4e00\u4e2a\u4ece\u03a9\u5230\u5177\u6709\u62d3\u6251\u7ed3\u6784\u7684\u72b6\u6001\u7a7a\u95f4\u7684 \u6ce2\u83b1\u5c14\u53ef\u6d4b\u51fd\u6570 Borel-measurable function \u3002<\/span><\/p>\n


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\u8003\u8651\u5230\u5728\u6bcf\u4e2a\u968f\u673a\u53d8\u91cf\u72b6\u6001\u7a7a\u95f4\u4e2d\u7684\u6ce2\u83b1\u5c14\u6d4b\u5ea6 Borel measure\uff08\u5173\u4e8e\u5f00\u653e\u96c6\u751f\u6210\u7684\u03c3\u4ee3\u6570\uff09<\/span><\/p>\n

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\u8fd9\u88ab\u79f0\u4e3a \u524d\u63a8\u6d4b\u5ea6 Pushforward measure<\/span><\/p>\n


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\u968f\u673a\u53d8\u91cf\u7684\u652f\u6491\u96c6\u5b9a\u4e49\u4e3a\u8be5\u6d4b\u5ea6\u7684\u62d3\u6251\u652f\u6491\u96c6\uff0c\u5373<\/span><\/p>\n


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\u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u7ed9\u5b9a\u5176\u4e2d\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u503c\uff08\u6216\u901a\u8fc7 \u79ef\u62d3\u6251 product topology \u83b7\u5f97\u66f4\u591a\uff09\u7684\u60c5\u51b5\u4e0b\u6b63\u5f0f\u5b9a\u4e49\u6761\u4ef6\u6982\u7387\u6d4b\u5ea6 Conditional probability distribution \u3002\u4ee4M\u4e3a\u03a9\u7684\u53ef\u6d4b\u5b50\u96c6\uff08\u5373M\u2208F)\uff0c\u4ee4x\u2208suppX<\/span>\u3002<\/span>\u7136\u540e\uff0c\u4f7f\u7528 \u5206\u89e3\u5b9a\u7406 Disintegration theorem \uff1a<\/span><\/p>\n


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\u5728x\u7684\u5f00\u653e\u90bb\u57dfU\u5904\u53d6\u6781\u9650\uff0c\u56e0\u4e3a\u76f8\u5bf9\u4e8e\u96c6\u5305\u542b Set inclusion\uff0c\u5b83\u4eec\u53ef\u4ee5\u4efb\u610f\u53d8\u5c0f\u3002<\/span><\/p>\n


\u6700\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7 \u52d2\u8d1d\u683c\u79ef\u5206 Lebesgue integration\u6765\u5b9a\u4e49\u6761\u4ef6\u4e92\u4fe1\u606f\uff1a<\/span><\/p>\n


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\u5176\u4e2d\u88ab\u79ef\u51fd\u6570\u662f\u62c9\u4e1c-\u5c3c\u79d1\u8fea\u59c6\u5bfc\u6570 Radon\u2013Nikodym derivative\u7684\u5bf9\u6570\uff0c\u6d89\u53ca\u6211\u4eec\u521a\u521a\u5b9a\u4e49\u7684\u4e00\u4e9b\u6761\u4ef6\u6982\u7387\u6d4b\u5ea6\u3002<\/span><\/p>\n


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\u5728\u8bf8\u5982I(A;B|C)\u7684\u8868\u8fbe\u5f0f\u4e2d\uff0cA,B\u548cC\u4e0d\u9650\u4e8e\u8868\u793a\u5355\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5b83\u4eec\u540c\u65f6\u53ef\u4ee5\u8868\u793a\u5728\u540c\u4e00\u6982\u7387\u7a7a\u95f4\u4e0a\u5b9a\u4e49\u7684\u4efb\u610f\u968f\u673a\u53d8\u91cf\u96c6\u5408\u7684\u8054\u5408\u5206\u5e03\u3002\u7c7b<\/span>\u4f3c\u6982\u7387\u8bba\u4e2d\u7684\u8868\u8fbe\u65b9\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9017\u53f7\u6765\u8868\u793a\u8fd9\u79cd\u8054\u5408\u5206\u5e03\uff0c\u4f8b\u5982I(A0,A1;B1,B2,B3|C0,C1)\u3002\u56e0\u6b64\uff0c\u4f7f\u7528\u5206\u53f7\uff08\u6216\u6709\u65f6\u7528\u5192\u53f7\u6216\u6954\u5f62\u2227\uff09\u6765\u5206\u9694\u4e92\u4fe1\u606f\u7b26\u53f7\u7684\u4e3b\u8981\u53c2\u6570\u3002\uff08\u5728\u8054\u5408\u71b5\u7684\u7b26\u53f7\u4e2d\uff0c\u4e0d\u9700\u8981\u4f5c\u8fd9\u6837\u7684\u533a\u5206\uff0c\u56e0\u4e3a\u4efb\u610f\u6570\u91cf\u968f\u673a\u53d8\u91cf\u7684 \u8054\u5408\u71b5 Joint entropy\u4e0e\u5b83\u4eec\u8054\u5408\u5206\u5e03\u7684\u71b5\u76f8\u540c\u3002\uff09<\/span>
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I(X;Y|Z)>=0\u3002<\/span><\/p>\n


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\u4f8b\u5982\u5176\u5dee\u503cI(X;Y)-I(X;Y|Z)\uff0c\u79f0\u4e3a \u4ea4\u4e92\u4fe1\u606f Interaction information (\u6ce8\u610f\u533a\u5206\u4e92\u4fe1\u606fMutual information)\uff0c\u53ef\u4ee5\u4e3a\u6b63\uff0c\u8d1f\u6216\u96f6\u3002<\/span>\u5373\u4f7f\u968f\u673a\u53d8\u91cf\u662f\u6210\u5bf9\u72ec\u7acb\u7684\u4e5f\u662f\u5982\u6b64\u3002<\/span>\u6bd4\u5982\u4ee5\u4e0b\u60c5\u51b5\u4e0b\uff1a<\/span>
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X\uff0cY\u548cZ\u662f\u6210\u5bf9\u72ec\u7acb\u7684\uff0c\u7279\u522b\u662fI(X;Y)=0\uff0c\u4e0d\u8fc7\u8fd9\u91ccI(X;Y|Z)=1\u3002<\/span><\/p>\n


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https:\/\/campus.swarma.org\/course\/3164<\/span><\/span><\/strong><\/p>\n

\u4fe1\u606f\u8bba\uff08information theory\uff09\u6d89\u53ca\u4fe1\u606f\u7684\u91cf\u5316\u3001\u5b58\u50a8\u548c\u901a\u4fe1\u7b49\u3002\u4fe1\u606f\u8bba\u662f\u7531\u514b\u52b3\u5fb7\u00b7\u9999\u519c\u53d1\u5c55\u6765\u7684\uff0c\u7528\u6765\u627e\u51fa\u4fe1\u53f7\u5904\u7406\u4e0e\u901a\u4fe1\u64cd\u4f5c\u7684\u57fa\u672c\u9650\u5236\uff0c\u5982\u6570\u636e\u538b\u7f29\u3001\u53ef\u9760\u7684\u5b58\u50a8\u548c\u6570\u636e\u4f20\u8f93\u7b49\u3002\u81ea\u521b\u7acb\u4ee5\u6765\uff0c\u5b83\u5df2\u62d3\u5c55\u5e94\u7528\u5230\u8bb8\u591a\u5176\u4ed6\u9886\u57df\uff0c\u5305\u62ec\u7edf\u8ba1\u63a8\u65ad\u3001\u5bc6\u7801\u5b66\u3001\u795e\u7ecf\u751f\u7269\u5b66\u3001\u8fdb\u5316\u8bba\u3001\u91cf\u5b50\u8ba1\u7b97\u3001\u527d\u7a83\u68c0\u6d4b\u548c\u5176\u4ed6\u5f62\u5f0f\u7684\u6570\u636e\u5206\u6790\u3002<\/span><\/p>\n


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