{"id":2166,"date":"2024-12-24T15:52:31","date_gmt":"2024-12-24T07:52:31","guid":{"rendered":"http:\/\/blog.xtaa.cn\/?p=2166"},"modified":"2024-12-24T15:52:31","modified_gmt":"2024-12-24T07:52:31","slug":"%e8%b4%9d%e5%8f%b6%e6%96%af%e5%ae%9a%e7%90%86%ef%bc%88%e5%8e%9f%e7%90%86%e7%af%87%ef%bc%89","status":"publish","type":"post","link":"http:\/\/blog.xtaa.cn\/index.php\/2024\/12\/24\/%e8%b4%9d%e5%8f%b6%e6%96%af%e5%ae%9a%e7%90%86%ef%bc%88%e5%8e%9f%e7%90%86%e7%af%87%ef%bc%89\/","title":{"rendered":"\u8d1d\u53f6\u65af\u5b9a\u7406\uff08\u539f\u7406\u7bc7\uff09"},"content":{"rendered":"\n<p>\u8f6c\u81ea<a href=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/index.html\">https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/index.html<\/a><br>\u53ef\u4ee5\u4e0d\u7528\u770b\u6587\u7ae0\uff0c\u76f4\u63a5\u770b\u6cb9\u7ba1\u4e0a\u89c6\u9891<a href=\"https:\/\/www.youtube.com\/watch?v=Pu675cHJ7bg\" target=\"_blank\" rel=\"noreferrer noopener\">Bayes&#8217; Theorem \u8d1d\u53f6\u65af\u5b9a\u7406<\/a><\/p>\n\n\n\n<p>\u6258\u9a6c\u65af\u00b7\u8d1d\u53f6\u65af\uff08Thomas Bayes\uff09\u662f18\u4e16\u7eaa\u7684\u82f1\u56fd\u6570\u5b66\u5bb6\uff0c\u4e5f\u662f\u4e00\u4f4d\u8654\u8bda\u7684\u7267\u5e08\u3002\u636e\u8bf4\u4ed6\u4e3a\u4e86\u53cd\u9a73\u5bf9\u4e0a\u5e1d\u7684\u8d28\u7591\u800c\u63a8\u5bfc\u51fa\u8d1d\u53f6\u65af\u5b9a\u7406\u3002\u8d1d\u53f6\u65af\u5b9a\u7406\u662f\u4e00\u4e2a\u7531\u7ed3\u679c\u5012\u63a8\u539f\u56e0\u7684\u6982\u7387\u7b97\u6cd5\uff0c\u5728\u8d1d\u53f6\u65af\u63d0\u51fa\u8fd9\u4e2a\u6761\u4ef6\u6982\u7387\u516c\u5f0f\u540e\uff0c\u5f88\u957f\u4e00\u6bb5\u65f6\u95f4\uff0c\u5927\u5bb6\u5e76\u6ca1\u6709\u89c9\u5f97\u5b83\u6709\u4ec0\u4e48\u4f5c\u7528\uff0c\u5e76\u4e00\u76f4\u53d7\u5230\u4e3b\u6d41\u7edf\u8ba1\u5b66\u6d3e\u7684\u6392\u65a5\u3002\u76f4\u5230\u8ba1\u7b97\u673a\u8bde\u751f\u540e\uff0c\u4eba\u4eec\u53d1\u73b0\uff0c\u8d1d\u53f6\u65af\u5b9a\u7406\u53ef\u4ee5\u5e7f\u6cdb\u5e94\u7528\u5728\u6570\u636e\u5206\u6790\u3001\u6a21\u5f0f\u8bc6\u522b\u3001\u7edf\u8ba1\u51b3\u7b56\uff0c\u4ee5\u53ca\u6700\u706b\u7684\u4eba\u5de5\u667a\u80fd\u4e2d\uff0c\u7ed3\u679c\uff0c\u8d1d\u53f6\u65af\u5b9a\u7406\u662f\u5982\u6b64\u6709\u7528\uff0c\u4ee5\u81f3\u4e8e\u4e0d\u4ec5\u5e94\u7528\u5728\u8ba1\u7b97\u673a\u4e0a\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u5728\u7ecf\u6d4e\u5b66\u3001\u5fc3\u7406\u5b66\u3001\u535a\u5f08\u8bba\u7b49\u5404\u79cd\u9886\u57df\uff0c\u53ef\u4ee5\u8bf4\uff0c\u638c\u63e1\u5e76\u5e94\u7528\u8d1d\u53f6\u65af\u5b9a\u7406\uff0c\u662f\u6bcf\u4e2a\u4eba\u5fc5\u5907\u7684\u6280\u80fd\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u63a8\u8350\u4e24\u4e2a\u89c6\u9891\uff0c\u6df1\u5165\u6d45\u51fa\u5730\u89e3\u91ca\u4e86\u8d1d\u53f6\u65af\u5b9a\u7406\uff1a<\/p>\n\n\n\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Pu675cHJ7bg\" target=\"_blank\" rel=\"noreferrer noopener\">Bayes&#8217; Theorem \u8d1d\u53f6\u65af\u5b9a\u7406<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HZGCoVF3YvM\" target=\"_blank\" rel=\"noreferrer noopener\">Bayes theorem, the geometry of changing beliefs<\/a><\/p>\n\n\n\n<p>\u5982\u679c\u4f60\u4e0d\u60f3\u82b1\u592a\u591a\u65f6\u95f4\u770b\u89c6\u9891\uff0c\u53ef\u4ee5\u7ee7\u7eed\u9605\u8bfb\uff0c\u6211\u628a\u89c6\u9891\u5185\u5bb9\u7f16\u8bd1\u6210\u6587\u5b57\uff0c\u4ee5\u4fbf\u5feb\u901f\u5b66\u4e60\u8d1d\u53f6\u65af\u5b9a\u7406\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u641e\u660e\u767d\u8d1d\u53f6\u65af\u5b9a\u7406\u7a76\u7adf\u8981\u89e3\u51b3\u4ec0\u4e48\u95ee\u9898\uff0c\u6211\u4eec\u5148\u770b\u4e00\u4e2a\u73b0\u5b9e\u751f\u6d3b\u7684\u4f8b\u5b50\uff1a<\/p>\n\n\n\n<p>\u5df2\u77e5\u6709\u4e00\u79cd\u75be\u75c5\uff0c\u53d1\u75c5\u7387\u662f0.1%\u3002\u9488\u5bf9\u8fd9\u79cd\u75be\u75c5\u7684\u6d4b\u8bd5\u975e\u5e38\u51c6\u786e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c\u6709\u75c5\uff0c\u5219\u51c6\u786e\u7387\u662f99%\uff08\u5373\u67091%\u672a\u68c0\u51fa\u9633\u6027\uff09\uff1b<\/li>\n\n\n\n<li>\u5982\u679c\u6ca1\u6709\u75c5\uff0c\u5219\u8bef\u62a5\u7387\u662f2%\uff08\u5373\u67092%\u8bef\u62a5\u4e3a\u9633\u6027\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u73b0\u5728\uff0c\u5982\u679c\u4e00\u4e2a\u4eba\u6d4b\u8bd5\u663e\u793a\u9633\u6027\uff0c\u8bf7\u95ee\u4ed6\u60a3\u75c5\u7684\u6982\u7387\u662f\u591a\u5c11\uff1f<\/p>\n\n\n\n<p>\u5982\u679c\u6211\u4eec\u4ece\u5927\u8857\u4e0a\u968f\u4fbf\u627e\u4e00\u4e2a\u4eba\uff0c\u90a3\u4e48\u4ed6\u60a3\u75c5\u7684\u6982\u7387\u5c31\u662f0.1%\uff0c\u56e0\u4e3a\u8fd9\u4e2a\u6982\u7387\u662f\u57fa\u4e8e\u5386\u53f2\u7edf\u8ba1\u6570\u636e\u7684\u5148\u9a8c\u6982\u7387\u3002<\/p>\n\n\n\n<p>\u73b0\u5728\uff0c\u4ed6\u505a\u4e86\u4e00\u6b21\u6d4b\u8bd5\uff0c\u7ed3\u679c\u4e3a\u9633\u6027\uff0c\u6211\u4eec\u8981\u8ba1\u7b97\u4ed6\u60a3\u75c5\u7684\u6982\u7387\uff0c\u5c31\u662f\u8ba1\u7b97\u6761\u4ef6\u6982\u7387\uff0c\u5373\uff1a\u5728\u6d4b\u8bd5\u4e3a\u9633\u6027\u8fd9\u4e00\u6761\u4ef6\u4e0b\uff0c\u60a3\u75c5\u7684\u6982\u7387\u662f\u591a\u5c11\u3002<\/p>\n\n\n\n<p>\u4ece\u76f4\u89c9\u4e0a\u8fd9\u4e2a\u4eba\u60a3\u75c5\u7684\u6982\u7387\u5927\u4e8e0.1%\uff0c\u4f46\u4e5f\u80af\u5b9a\u5c0f\u4e8e99%\u3002\u7a76\u7adf\u662f\u591a\u5c11\uff0c\u600e\u4e48\u8ba1\u7b97\uff0c\u6211\u4eec\u5148\u653e\u4e00\u653e\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u7406\u89e3\u6761\u4ef6\u6982\u7387\uff0c\u6211\u4eec\u6362\u4e00\u4e2a\u66f4\u7b80\u5355\u7684\u4f8b\u5b50\uff1a\u63b7\u4e24\u6b21\u9ab0\u5b50\uff0c\u4e00\u5171\u53ef\u80fd\u51fa\u73b0\u7684\u7ed3\u679c\u67096&#215;6=36\u79cd\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/sample-all.png\" alt=\"sample space\"\/><\/figure>\n\n\n\n<p>\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u6837\u672c\u7a7a\u95f4\uff0c\u6bcf\u4e2a\u6837\u672c\u7684\u6982\u7387\u5747\u4e3a1\/36\uff0c\u8fd9\u4e2a\u5f88\u597d\u7406\u89e3\u3002<\/p>\n\n\n\n<p>\u5982\u679c\u6211\u4eec\u5b9a\u4e49\u4e8b\u4ef6A\u4e3a\uff1a\u81f3\u5c11\u6709\u4e00\u4e2a\u9ab0\u5b50\u662f2\uff0c\u90a3\u4e48\u4e8b\u4ef6A\u7684\u6837\u672c\u7a7a\u95f4\u5982\u4e0b\u56fe\u7ea2\u8272\u90e8\u5206\u6240\u793a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/sample-a.png\" alt=\"Event A\"\/><\/figure>\n\n\n\n<p>\u4e8b\u4ef6A\u4e00\u5171\u670911\u79cd\u60c5\u51b5\uff0c\u6211\u4eec\u8ba1\u7b97\u4e8b\u4ef6A\u7684\u6982\u7387P(A)\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/pa.png\" alt=\"P(A)\"\/><\/figure>\n\n\n\n<p>\u6211\u4eec\u518d\u5b9a\u4e49\u4e8b\u4ef6B\uff1a\u4e24\u4e2a\u9ab0\u5b50\u4e4b\u548c\u4e3a7\uff0c\u90a3\u4e48\u4e8b\u4ef6B\u7684\u6837\u672c\u7a7a\u95f4\u5982\u4e0b\u56fe\u7eff\u8272\u90e8\u5206\u6240\u793a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/sample-b.png\" alt=\"Event B\"\/><\/figure>\n\n\n\n<p>\u4e8b\u4ef6B\u4e00\u5171\u67096\u79cd\u60c5\u51b5\uff0c\u6211\u4eec\u8ba1\u7b97\u4e8b\u4ef6B\u7684\u6982\u7387P(B)\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/pb.png\" alt=\"P(B)\"\/><\/figure>\n\n\n\n<p>\u63a5\u4e0b\u6765\u6211\u4eec\u7528P(A\u2229B)\u8868\u793aA\u548cB\u540c\u65f6\u53d1\u751f\u7684\u6982\u7387\uff0cA\u2229B\u5c31\u662fA\u548cB\u7684\u4ea4\u96c6\uff0c\u5982\u4e0b\u56fe\u84dd\u8272\u90e8\u5206\u6240\u793a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/sample-p-a-and-b.png\" alt=\"P(A\u2229B)\"\/><\/figure>\n\n\n\n<p>\u663e\u7136A\u2229B\u53ea\u6709\u4e24\u79cd\u60c5\u51b5\uff0c\u56e0\u6b64\uff0c\u8ba1\u7b97P(A\u2229B)\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/p-a-and-b.png\" alt=\"P(A\u2229B)\"\/><\/figure>\n\n\n\n<p>\u63a5\u4e0b\u6765\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba8\u8bba\u6761\u4ef6\u6982\u7387\u4e86\u3002\u6211\u4eec\u7528P(A|B)\u8868\u793a\u5728B\u53d1\u751f\u7684\u6761\u4ef6\u4e0b\uff0cA\u53d1\u751f\u7684\u6982\u7387\u3002\u7531\u4e8eB\u5df2\u7ecf\u53d1\u751f\uff0c\u6240\u4ee5\uff0c\u6837\u672c\u7a7a\u95f4\u5c31\u662fB\u7684\u6837\u672c\u6570\u91cf6\uff0c\u800c\u8981\u53d1\u751fA\u5219\u53ea\u80fd\u662fA\u3001B\u540c\u65f6\u53d1\u751f\uff0c\u5373A\u2229B\uff0c\u6709\u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n\n\n\n<p>\u56e0\u6b64\uff0c\u8ba1\u7b97P(A|B)\u5982\u4e0b\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/p-a-b.png\" alt=\"P(A|B)\"\/><\/figure>\n\n\n\n<p>\u540c\u7406\uff0c\u6211\u4eec\u7528P(B|A)\u8868\u793a\u5728A\u53d1\u751f\u7684\u6761\u4ef6\u4e0b\uff0cB\u53d1\u751f\u7684\u6982\u7387\u3002\u6b64\u65f6\uff0c\u5206\u5b50\u4ecd\u7136\u662fA\u2229B\u7684\u6837\u672c\u6570\u91cf\uff0c\u4f46\u5206\u6bcd\u53d8\u6210A\u7684\u6837\u672c\u6570\u91cf\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/p-b-a.png\" alt=\"P(B|A)\"\/><\/figure>\n\n\n\n<p>\u53ef\u89c1\uff0c\u6761\u4ef6\u6982\u7387P(A|B)\u548cP(B|A)\u662f\u4e0d\u540c\u7684\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u518d\u56de\u5230A\u3001B\u540c\u65f6\u53d1\u751f\u7684\u6982\u7387\uff0c\u89c2\u5bdfP(A\u2229B)\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/p-a-and-b-2.png\" alt=\"P(B|A)xP(A)\"\/><\/figure>\n\n\n\n<p>\u540c\u7406\uff0cP(A\u2229B)\u8fd8\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/p-a-and-b-3.png\" alt=\"P(A|B)xP(B)\"\/><\/figure>\n\n\n\n<p>\u56e0\u6b64\uff0c\u6839\u636e\u4e0a\u8ff0\u4e24\u4e2a\u7b49\u5f0f\uff0c\u6211\u4eec\u63a8\u5bfc\u51fa\u4e0b\u9762\u7684\u7b49\u5f0f\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>A<\/em>\u2229<em>B<\/em>)=<em>P<\/em>(<em>A<\/em>\u2223<em>B<\/em>)\u00d7<em>P<\/em>(<em>B<\/em>)=<em>P<\/em>(<em>B<\/em>\u2223<em>A<\/em>)\u00d7<em>P<\/em>(<em>A<\/em>)<\/p>\n\n\n\n<p>\u628a\u5de6\u8fb9\u7684<code>P(A\u2229B)<\/code>\u53bb\u6389\uff0c\u6211\u4eec\u5f97\u5230\u7b49\u5f0f\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>A<\/em>\u2223<em>B<\/em>)\u00d7<em>P<\/em>(<em>B<\/em>)=<em>P<\/em>(<em>B<\/em>\u2223<em>A<\/em>)\u00d7<em>P<\/em>(<em>A<\/em>)<\/p>\n\n\n\n<p>\u6700\u540e\uff0c\u6574\u7406\u4e00\u4e0b\u7b49\u5f0f\uff0c\u6211\u4eec\u63a8\u5bfc\u51fa\u8d1d\u53f6\u65af\u5b9a\u7406\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>A<\/em>\u2223<em>B<\/em>)=<em>P<\/em>(<em>B<\/em>)<em>P<\/em>(<em>B<\/em>\u2223<em>A<\/em>)\u00d7<em>P<\/em>(<em>A<\/em>)\u200b<\/p>\n\n\n\n<p>\u8fd9\u5c31\u662f\u8457\u540d\u7684\u8d1d\u53f6\u65af\u5b9a\u7406\uff0c\u5b83\u8868\u793a\uff0c\u5f53\u51fa\u73b0B\u65f6\uff0c\u5982\u4f55\u8ba1\u7b97A\u7684\u6982\u7387\u3002<\/p>\n\n\n\n<p>\u5f88\u591a\u65f6\u5019\uff0c\u6211\u4eec\u628a<code>A<\/code>\u6539\u5199\u4e3a<code>H<\/code>\uff0c\u628a<code>B<\/code>\u6539\u5199\u4e3a<code>E<\/code>\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b<\/p>\n\n\n\n<p>H\u8868\u793aHypothesis\uff08\u5047\u8bbe\uff09\uff0cE\u8868\u793aEvidence\uff08\u8bc1\u636e\uff09\uff0c\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u610f\u4e49\u5c31\u5728\u4e8e\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5148\u9a8c\u6982\u7387P(H)\uff0c\u5728\u51fa\u73b0\u4e86\u8bc1\u636eE\u7684\u60c5\u51b5\u4e0b\uff0c\u8ba1\u7b97\u540e\u9a8c\u6982\u7387P(H|E)\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u8ba1\u7b97<\/h3>\n\n\n\n<p>\u6709\u4e86\u8d1d\u53f6\u65af\u5b9a\u7406\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u56de\u5230\u5f00\u5934\u7684\u95ee\u9898\uff1a<\/p>\n\n\n\n<p>\u5df2\u77e5\u6709\u4e00\u79cd\u75be\u75c5\uff0c\u53d1\u75c5\u7387\u662f0.1%\u3002\u9488\u5bf9\u8fd9\u79cd\u75be\u75c5\u7684\u6d4b\u8bd5\u975e\u5e38\u51c6\u786e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c\u6709\u75c5\uff0c\u5219\u51c6\u786e\u7387\u662f99%\uff08\u5373\u67091%\u672a\u68c0\u51fa\u9633\u6027\uff09\uff1b<\/li>\n\n\n\n<li>\u5982\u679c\u6ca1\u6709\u75c5\uff0c\u5219\u8bef\u62a5\u7387\u662f2%\uff08\u5373\u67092%\u8bef\u62a5\u4e3a\u9633\u6027\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u73b0\u5728\uff0c\u5982\u679c\u4e00\u4e2a\u4eba\u6d4b\u8bd5\u663e\u793a\u9633\u6027\uff0c\u8bf7\u95ee\u4ed6\u60a3\u75c5\u7684\u6982\u7387\u662f\u591a\u5c11\uff1f<\/p>\n\n\n\n<p>\u7528H\u8868\u793a\u60a3\u75c5\uff0cE\u8868\u793a\u6d4b\u8bd5\u4e3a\u9633\u6027\uff0c\u90a3\u4e48\uff0c\u6211\u4eec\u8981\u8ba1\u7b97\u5728\u6d4b\u8bd5\u4e3a\u9633\u6027\u7684\u6761\u4ef6\u4e0b\uff0c\u4e00\u4e2a\u4eba\u60a3\u75c5\u7684\u6982\u7387\uff0c\u5c31\u662f\u8ba1\u7b97P(H|E)\u3002\u6839\u636e\u8d1d\u53f6\u65af\u5b9a\u7406\uff0c\u8ba1\u7b97\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b<\/p>\n\n\n\n<p>P(H)\u8868\u793a\u60a3\u75c5\u7684\u6982\u7387\uff0c\u6839\u636e\u53d1\u75c5\u7387\u53ef\u77e5\uff0cP(H)=0.1%\uff1b<\/p>\n\n\n\n<p>P(E|H)\u8868\u793a\u5728\u60a3\u75c5\u7684\u60c5\u51b5\u4e0b\uff0c\u6d4b\u8bd5\u4e3a\u9633\u6027\u7684\u6982\u7387\uff0c\u6839\u636e\u201c\u5982\u679c\u6709\u75c5\uff0c\u5219\u51c6\u786e\u7387\u662f99%\u201d\u53ef\u77e5\uff0cP(E|H)=99%\uff1b<\/p>\n\n\n\n<p>P(E)\u8868\u793a\u6d4b\u8bd5\u4e3a\u9633\u6027\u7684\u6982\u7387\u3002\u8fd9\u4e2a\u6982\u7387\u5c31\u7a0d\u5fae\u590d\u6742\u70b9\uff0c\u56e0\u4e3a\u5b83\u662f\u6307\u5bf9\u6240\u6709\u4eba\uff08\u5305\u542b\u75c5\u4eba\u548c\u5065\u5eb7\u4eba\uff09\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7ed3\u679c\u9633\u6027\u7684\u6982\u7387\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u628a\u68c0\u6d4b\u4eba\u6570\u653e\u5927\uff0c\u4f8b\u5982\u653e\u5927\u523010\u4e07\u4eba\uff0c\u5bf910\u4e07\u4eba\u8fdb\u884c\u68c0\u6d4b\uff0c\u6839\u636e\u53d1\u75c5\u7387\u53ef\u77e5\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6709100\u4eba\u662f\u75c5\u4eba\uff0c\u53e6\u591699900\u662f\u5065\u5eb7\u4eba\uff1b<\/li>\n\n\n\n<li>\u5bf9100\u4e2a\u75c5\u4eba\u8fdb\u884c\u6d4b\u8bd5\uff0c\u670999\u4eba\u663e\u793a\u9633\u6027\uff0c\u53e6\u67091\u4eba\u672a\u68c0\u51fa\uff08\u9634\u6027\uff09\uff1b<\/li>\n\n\n\n<li>\u5bf999900\u4e2a\u5065\u5eb7\u4eba\u8fdb\u884c\u6d4b\u8bd5\uff0c\u67092%=1998\u4eba\u663e\u793a\u9633\u6027\uff08\u8bef\u62a5\uff09\uff0c\u53e6\u670998%=97902\u4eba\u4e3a\u9634\u6027\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u4e0b\u56fe\u663e\u793a\u4e86\u68c0\u6d4b\u4e3a\u9633\u6027\u7684\u7ed3\u679c\u7684\u5206\u5e03\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>           \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n           \u2502100000 \u2502\n           \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n               \u2502\n       \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n       \u25bc               \u25bc\n   \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510       \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n   \u2502  100  \u2502       \u2502 99900 \u2502\n   \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518       \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n       \u2502               \u2502\n   \u250c\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2510       \u250c\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2510\n   \u25bc       \u25bc       \u25bc       \u25bc\n\u250c\u2500\u2500\u2500\u2500\u2500\u2510 \u250c\u2500\u2500\u2500\u2500\u2500\u2510 \u250c\u2500\u2500\u2500\u2500\u2500\u2510 \u250c\u2500\u2500\u2500\u2500\u2500\u2510\n\u2502 99  \u2502 \u2502  1  \u2502 \u25021998 \u2502 \u250297902\u2502\n\u2514\u2500\u2500\u2500\u2500\u2500\u2518 \u2514\u2500\u2500\u2500\u2500\u2500\u2518 \u2514\u2500\u2500\u2500\u2500\u2500\u2518 \u2514\u2500\u2500\u2500\u2500\u2500\u2518\n   \u2502               \u2502\n   \u25bc               \u25bc\n   +               +\n<\/code><\/pre>\n\n\n\n<p>\u6240\u4ee5\uff0c\u5bf9\u4e8e10\u4e07\u4eba\u7684\u6837\u672c\u7a7a\u95f4\u6765\u8bf4\uff0c\u4e8b\u4ef6E=\u663e\u793a\u9633\u6027\u7684\u6982\u7387\u4e3a(99+1998)\/100000=2.097%\u3002<\/p>\n\n\n\n<p>\u5e26\u5165\u8d1d\u53f6\u65af\u5b9a\u7406\uff0c\u8ba1\u7b97P(H|E)\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b=2.097%99%\u00d70.1%\u200b=0.020970.99\u00d70.001\u200b=0.04721=4.721%<\/p>\n\n\n\n<p>\u8ba1\u7b97\u7ed3\u679c\u4e3a\u60a3\u75c5\u7684\u6982\u7387\u4e3a4.721%\uff0c\u8fd9\u4e2a\u6982\u7387\u8fdc\u5c0f\u4e8e99%\uff0c\u4e14\u4e0e\u5927\u591a\u6570\u4eba\u7684\u76f4\u89c9\u4e0d\u540c\uff0c\u539f\u56e0\u5728\u4e8e\u5e9e\u5927\u7684\u5065\u5eb7\u4eba\u7fa4\u5bfc\u81f4\u7684\u8bef\u62a5\u6570\u91cf\u8fdc\u591a\u4e8e\u75c5\u4eba\uff0c\u5f53\u51fa\u73b0\u201c\u68c0\u6d4b\u9633\u6027\u201d\u7684\u8bc1\u636e\u65f6\uff0c\u60a3\u75c5\u7684\u6982\u7387\u4ece\u5148\u9a8c\u6982\u73870.1%\u63d0\u5347\u52304.721%\uff0c\u8fd8\u8fdc\u4e0d\u8db3\u4ee5\u786e\u8bca\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u8868\u793a<\/h3>\n\n\n\n<p>\u5728\u4e0a\u8ff0\u8ba1\u7b97\u4e2d\uff0c\u6211\u4eec\u53d1\u73b0\u8ba1\u7b97P(E)\u662f\u6bd4\u8f83\u56f0\u96be\u7684\uff0c\u5f88\u591a\u65f6\u5019\uff0c\u751a\u81f3\u65e0\u6cd5\u77e5\u9053P(E)\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u9700\u8981\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u8868\u793a\u5f62\u5f0f\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u7528P(H)\u8868\u793aH\u53d1\u751f\u7684\u6982\u7387\uff0c\u7528H\u8868\u793aH\u4e0d\u53d1\u751f\uff0cP(H)\u8868\u793aH\u4e0d\u53d1\u751f\u7684\u6982\u7387\u3002\u663e\u7136P(H)=1-P(H)\u3002<\/p>\n\n\n\n<p>\u4e0b\u56fe\u7ea2\u8272\u90e8\u5206\u8868\u793aH\uff0c\u7ea2\u8272\u90e8\u5206\u4ee5\u5916\u5219\u8868\u793aH\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/ph.png\" alt=\"P(H)\"\/><\/figure>\n\n\n\n<p>\u4e8b\u4ef6E\u7528\u7eff\u8272\u8868\u793a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/liaoxuefeng.com\/blogs\/all\/2023-08-27-bayes-explain\/pe.png\" alt=\"P(E)\"\/><\/figure>\n\n\n\n<p>\u53ef\u89c1\uff0cP(E)\u53ef\u4ee5\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u4e00\u90e8\u5206\u662fE\u548cH\u7684\u4ea4\u96c6\uff0c\u53e6\u4e00\u90e8\u5206\u662fE\u548cH\u7684\u4ea4\u96c6\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>\u2229<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2229<em>H<\/em>)<\/p>\n\n\n\n<p>\u6839\u636e\u4e0a\u6587\u7684\u516c\u5f0fP(A\u2229B)=P(A|B)xP(B)\uff0c\u4ee3\u5165\u53ef\u5f97\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>\u2229<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2229<em>H<\/em>)=<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)<\/p>\n\n\n\n<p>\u628aP(E)\u66ff\u6362\u6389\uff0c\u6211\u4eec\u5f97\u5230\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u5199\u6cd5\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b<\/p>\n\n\n\n<p>\u7528\u8fd9\u4e2a\u516c\u5f0f\u6765\u8ba1\u7b97\uff0c\u6211\u4eec\u5c31\u4e0d\u5fc5\u8ba1\u7b97P(E)\u4e86\u3002\u518d\u6b21\u56de\u5230\u5f00\u5934\u7684\u95ee\u9898\uff1a<\/p>\n\n\n\n<p>\u5df2\u77e5\u6709\u4e00\u79cd\u75be\u75c5\uff0c\u53d1\u75c5\u7387\u662f0.1%\u3002\u9488\u5bf9\u8fd9\u79cd\u75be\u75c5\u7684\u6d4b\u8bd5\u975e\u5e38\u51c6\u786e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c\u6709\u75c5\uff0c\u5219\u51c6\u786e\u7387\u662f99%\uff08\u5373\u67091%\u672a\u68c0\u51fa\u9633\u6027\uff09\uff1b<\/li>\n\n\n\n<li>\u5982\u679c\u6ca1\u6709\u75c5\uff0c\u5219\u8bef\u62a5\u7387\u662f2%\uff08\u5373\u67092%\u8bef\u62a5\u4e3a\u9633\u6027\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u73b0\u5728\uff0c\u5982\u679c\u4e00\u4e2a\u4eba\u6d4b\u8bd5\u663e\u793a\u9633\u6027\uff0c\u8bf7\u95ee\u4ed6\u60a3\u75c5\u7684\u6982\u7387\u662f\u591a\u5c11\uff1f<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(E|H)\u8868\u793a\u60a3\u75c5\u65f6\u68c0\u6d4b\u9633\u6027\u7684\u6982\u7387=99%\uff1b<\/li>\n\n\n\n<li>P(H)\u8868\u793a\u60a3\u75c5\u7684\u6982\u7387=0.1%\uff1b<\/li>\n\n\n\n<li>P(E|H)\u8868\u793a\u6ca1\u6709\u60a3\u75c5\u4f46\u68c0\u6d4b\u9633\u6027\u7684\u6982\u7387=2%\uff1b<\/li>\n\n\n\n<li>P(H)\u8868\u793a\u6ca1\u6709\u60a3\u75c5\u7684\u6982\u7387=1-P(H)=99.9%\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u4ee3\u5165\u516c\u5f0f\uff0c\u8ba1\u7b97\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b=99%\u00d70.1%+2%\u00d799.9%99%\u00d70.1%\u200b=0.04721=4.721%<\/p>\n\n\n\n<p>\u68c0\u6d4b\u4e3a\u9633\u6027\u8fd9\u4e00\u8bc1\u636e\u4f7f\u5f97\u60a3\u75c5\u7684\u6982\u7387\u4ece0.1%\u63d0\u5347\u52304.721%\u3002\u5047\u8bbe\u8fd9\u4e2a\u4eba\u53c8\u505a\u4e86\u4e00\u6b21\u68c0\u6d4b\uff0c\u7ed3\u679c\u4ecd\u7136\u662f\u9633\u6027\uff0c\u90a3\u4e48\u4ed6\u60a3\u75c5\u7684\u6982\u7387\u662f\u591a\u5c11\uff1f<\/p>\n\n\n\n<p>\u6211\u4eec\u4ecd\u7136\u4f7f\u7528\u8d1d\u53f6\u65af\u5b9a\u7406\u8ba1\u7b97\uff0c\u53ea\u4e0d\u8fc7\u73b0\u5728\u5148\u9a8c\u6982\u7387P(H)\u4e0d\u518d\u662f0.1%\uff0c\u800c\u662f4.721%\uff0cP(E|H)\u548cP(E|H)\u4ecd\u4fdd\u6301\u4e0d\u53d8\uff0c\u8ba1\u7b97\u65b0\u7684P(H|E)\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)+<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b=99%\u00d74.721%+2%\u00d7(1\u22124.721%)99%\u00d74.721%\u200b=0.71=71%<\/p>\n\n\n\n<p>\u7ed3\u679c\u4e3a71%\uff0c\u4e24\u6b21\u68c0\u6d4b\u4e3a\u9633\u6027\u7684\u7ed3\u679c\u4f7f\u5f97\u5148\u9a8c\u6982\u7387\u4ece0.1%\u63d0\u5347\u52304.721%\u518d\u63d0\u5347\u523071%\uff0c\u7ee7\u7eed\u7b2c\u4e09\u6b21\u68c0\u6d4b\u5982\u679c\u4e3a\u9633\u6027\u5219\u6982\u7387\u5c06\u63d0\u5347\u81f399.18%\u3002<\/p>\n\n\n\n<p>\u53ef\u89c1\uff0c\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u6838\u5fc3\u601d\u60f3\u5c31\u662f\u4e0d\u65ad\u6839\u636e\u65b0\u7684\u8bc1\u636e\uff0c\u5c06\u5148\u9a8c\u6982\u7387\u8c03\u6574\u4e3a\u540e\u9a8c\u6982\u7387\uff0c\u4f7f\u4e4b\u66f4\u63a5\u8fd1\u5ba2\u89c2\u4e8b\u5b9e\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5173\u4e8e\u4fe1\u5ff5<\/h3>\n\n\n\n<p>\u6211\u4eec\u518d\u56de\u987e\u4e00\u4e0b\u8d1d\u53f6\u65af\u5b9a\u7406\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>E<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u00d7<em>P<\/em>(<em>H<\/em>)\u200b<\/p>\n\n\n\n<p>\u7a0d\u5fae\u6539\u4e00\u4e0b\uff0c\u53d8\u4e3a\uff1a<\/p>\n\n\n\n<p><em>P<\/em>(<em>H<\/em>\u2223<em>E<\/em>)=<em>P<\/em>(<em>H<\/em>)\u00d7<em>P<\/em>(<em>E<\/em>)<em>P<\/em>(<em>E<\/em>\u2223<em>H<\/em>)\u200b<\/p>\n\n\n\n<p>P(H)\u662f\u5148\u9a8c\u6982\u7387\uff0cP(H|E)\u662f\u540e\u9a8c\u6982\u7387\uff0cP(E|H)\/P(E)\u88ab\u79f0\u4e3a\u8c03\u6574\u56e0\u5b50\uff0c\u5148\u9a8c\u6982\u7387\u4e58\u4ee5\u8c03\u6574\u56e0\u5b50\u5c31\u5f97\u5230\u540e\u9a8c\u6982\u7387\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679cP(H)=0\uff0c\u5219P(H|E)=0\uff1b\u5982\u679cP(H)=1\uff0c\u5219P(E|H)=P(E)\uff0cP(H|E)=1\u3002<\/p>\n\n\n\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u5148\u9a8c\u6982\u7387\u4e3a0%\u6216100%\uff0c\u90a3\u4e48\uff0c\u65e0\u8bba\u51fa\u73b0\u4efb\u4f55\u8bc1\u636eE\uff0c\u90fd\u65e0\u6cd5\u6539\u53d8\u540e\u9a8c\u6982\u7387P(H|E)\u3002\u8fd9\u5bf9\u6211\u4eec\u770b\u5f85\u4e16\u754c\u7684\u8ba4\u77e5\u6709\u91cd\u5927\u6307\u5bfc\u610f\u4e49\uff0c\u56e0\u4e3a\u8d1d\u53f6\u65af\u6982\u7387\u7684\u672c\u8d28\u662f\u4fe1\u5ff5\uff0c\u901a\u8fc7\u4e00\u6b21\u6b21\u4e8b\u4ef6\uff0c\u6211\u4eec\u53ef\u80fd\u52a0\u5f3a\u67d0\u79cd\u4fe1\u5ff5\uff0c\u4e5f\u53ef\u80fd\u51cf\u5f31\u67d0\u79cd\u4fe1\u5ff5\uff0c\u4f46\u5982\u679c\u4fe1\u5ff5\u4fdd\u6301100%\u62160%\uff0c\u5219\u53ef\u4ee5\u505a\u5230\u5bf9\u5916\u754c\u8f93\u5165\u5b8c\u5168\u201c\u514d\u75ab\u201d\u3002<\/p>\n\n\n\n<p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5341\u5e74\u524d\u8bb8\u591a\u4eba\u90fd\u8ba4\u4e3a\u6bd4\u7279\u5e01\u662f\u5e9e\u6c0f\u9a97\u5c40\uff0c\u5982\u679c100%\u575a\u5b9a\u5730\u6301\u6709\u8fd9\u79cd\u4fe1\u5ff5\uff0c\u90a3\u4e48\u4ed6\u5c06\u65e0\u89c6\u7528\u6237\u8d8a\u6765\u8d8a\u591a\u3001\u4ef7\u683c\u4e0a\u6da8\u3001\u4ea4\u6613\u91cf\u6269\u5927\u3001\u673a\u6784\u5165\u5e02\u7b49\u8bf8\u591a\u8bc1\u636e\uff0c\u81f3\u4eca\u4ecd\u7136\u4f1a\u575a\u4fe1\u6bd4\u7279\u5e01\u662f\u9a97\u5c40\u800c\u9519\u8fc7\u65e0\u6570\u6b21\u673a\u4f1a\u3002\uff08\u6ce8\uff1a\u6b64\u5904\u793a\u4f8b\u4e0d\u6784\u6210\u4efb\u4f55\u6295\u8d44\u5efa\u8bae\uff09<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u65b0\u751f\u4e8b\u7269\uff0c\u6bcf\u4e2a\u4eba\u90fd\u53ef\u4ee5\u6709\u975e\u5e38\u4e3b\u89c2\u7684\u5148\u9a8c\u6982\u7387\uff0c\u4f46\u53ea\u8981\u6211\u4eec\u4e0d\u628a\u5148\u9a8c\u6982\u7387\u5b9a\u6b7b\u4e3a0\u6216100%\uff0c\u5c31\u6709\u673a\u4f1a\u6539\u53d8\u81ea\u5df1\u7684\u4fe1\u5ff5\uff0c\u4ece\u800c\u66f4\u6709\u53ef\u80fd\u63a5\u8fd1\u5ba2\u89c2\u4e8b\u5b9e\uff0c\u8fd9\u4e5f\u662f\u8d1d\u53f6\u65af\u5b9a\u7406\u7684\u7cbe\u9ad3\uff1a<\/p>\n\n\n\n<blockquote 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