{"id":8209,"date":"2024-04-23T21:01:01","date_gmt":"2024-04-23T13:01:01","guid":{"rendered":""},"modified":"2024-04-23T21:01:01","modified_gmt":"2024-04-23T13:01:01","slug":"ANM(Nonlinear causal discovery with additive noise models)","status":"publish","type":"post","link":"https:\/\/mushiming.com\/8209.html","title":{"rendered":"ANM(Nonlinear causal discovery with additive noise models)"},"content":{"rendered":"<p>\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u6211\u4eec\u8bb2\u4e86\u5f53\u6570\u636e\u80cc\u540e\u7684\u751f\u6210\u673a\u5236\u662f\u7ebf\u6027\u7684\u65f6\u5019\uff0c\u6211\u4eec\u5982\u4f55\u4ece\u6570\u636e\u4e2d\u6062\u590d\u51fa\u5176\u80cc\u540e\u7684\u56e0\u679c\u7ed3\u6784\u3002\u90a3\u4e48\u5f53\u6570\u636e\u80cc\u540e\u751f\u6210\u7684\u673a\u5236\u662f\u975e\u7ebf\u6027\u7684\u65f6\u5019\uff0c\u6211\u4eec\u8be5\u5982\u4f55\u4ece\u6570\u636e\u4e2d\u6062\u590d\u51fa\u5176\u80cc\u540e\u7684\u56e0\u679c\u7ed3\u6784\u5462\uff1f<\/p>\n<p>\u672c\u7bc7\u6587\u7ae0\u5206\u4eab\u7684\u8bba\u6587\u5c31\u662f\u63d0\u51fa\u4e86\u4e00\u4e2a\u65b9\u6cd5\uff0c\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u5047\u8bbe\u6761\u4ef6\u4e0b\u6062\u590d\u51fa\u6570\u636e\u80cc\u540e\u7684\u56e0\u679c\u7ed3\u6784\u3002<\/p>\n<p><strong>\u5728LiNGAM\u4e2d\uff0c\u6211\u4eec\u77e5\u9053\u6211\u4eec\u4e4b\u6240\u4ee5\u80fd\u591f\u8bc6\u522b\u56e0\u679c\u7ed3\u6784\uff0c\u5f97\u76ca\u4e8e\u566a\u58f0\u9879\uff08\u6270\u52a8\u9879\uff09\u7684\u975e\u9ad8\u65af\u6027\u6240\u5177\u6709\u7684\u975e\u5bf9\u79f0\u6027\uff0c\u90a3\u4e48\u540c\u7406\uff0c\u5bf9\u4e8e\u5177\u6709\u975e\u5bf9\u79f0\u6027\u7684\u975e\u7ebf\u6027\u51fd\u6570\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4ece\u6570\u636e\u4e2d\u6062\u590d\u51fa\u56e0\u679c\u7ed3\u6784\u5417\uff1f<\/strong><\/p>\n<p><strong>ANM\u7ed9\u51fa\u4e86\u8fd9\u4e2a\u95ee\u9898\u7684\u56de\u7b54\uff0c\u662f\u7684\uff0c\u975e\u7ebf\u6027\u4e0e\u975e\u9ad8\u65af\u5177\u6709\u975e\u5e38\u76f8\u4f3c\u7684\u4e00\u4e9b\u6027\u8d28\uff0c\u975e\u7ebf\u6027\u4e5f\u53ef\u4ee5\u6253\u7834\u53d8\u91cf\u4e4b\u95f4\u7684\u5bf9\u79f0\u6027\uff0c\u8fdb\u800c\u5141\u8bb8\u6211\u4eec\u8bc6\u522b\u53d8\u91cf\u95f4\u7684\u56e0\u679c\u5173\u7cfb\u65b9\u5411\u3002<\/strong><\/p>\n<p><strong>\u8fd9\u7bc7\u6587\u7ae0\u6307\u51fa\uff1a\u5bf9\u4e8e\u4efb\u4f55\u5177\u6709\u52a0\u6027\u566a\u58f0\u7684\u975e\u7ebf\u6027\u6a21\u578b\uff0c\u51e0\u4e4e\u4efb\u4f55\u975e\u7ebf\u6027\uff08\u53ef\u9006\u6216\u4e0d\u53ef\u9006\uff09\u901a\u5e38\u90fd\u4f1a\u4ea7\u751f\u53ef\u8bc6\u522b\u7684\u6a21\u578b\u3002<\/strong>(for nonlinear models with additive noise almost any nonlinearities (invertible or not) will typically yield identifiable models)<\/p>\n<p>\u8fd9\u7bc7\u6587\u7ae0\u5047\u8bbe\u6570\u636e\u7684\u751f\u6210\u65b9\u5f0f\u4e3a\uff1a<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"x_i:=f_{i}(\\mathbf{x_{pa(i)}})+n_{i}\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c1\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c1\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff0c<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"f_i\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c2\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c2\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\u662f\u4efb\u610f\u51fd\u6570\uff0c<strong><img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"x_{pa(i)}\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c3\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c3\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/strong>\u4ee3\u8868\u7684\u5c31\u662f\u8282\u70b9i\u7684\u7236\u8282\u70b9\u96c6\u5408\u3002<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"n_i\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c4\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c4\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\u8868\u793a\u5177\u6709\u4efb\u610f\u6982\u7387\u5bc6\u5ea6\u7684\u566a\u58f0\u53d8\u91cf\u3002<\/p>\n<p>\u5728\u8fd9\u7bc7\u6587\u7ae0\u4e2d\uff0c\u901a\u8fc7\u6539\u53d8\u51fd\u6570f\u7684\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\uff0c\u8fdb\u800c\u8bf4\u660e\u4e86\u5f53\u51fd\u6570\u65f6\u975e\u7ebf\u6027\u65f6\uff0c\u6a21\u578b\u7684\u53ef\u8bc6\u522b\u6027\u3002\u5bf9\u6bd4\u56fe\u5982\u4e0b\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/a876d8b1463842f7a1d74154875d72f2.png\" =\"\" =\"\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c5\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c5\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<p>&nbsp;(a-c)\u662ff\u5c5e\u4e8e\u7ebf\u6027\u7684\u60c5\u51b5\uff0c\u7531\u4e0a\u56fe\u53ef\u77e5\uff0c\u5728\u56e0\u679c\u51fd\u6570\u662f\u7ebf\u6027\u4e14\u566a\u58f0\u9879\u65f6\u9ad8\u65af\u7684\u60c5\u51b5\u4e0b\uff0cP(x|y)\u4e0eP(y|x)\u7684\u503c\u5bf9\u4e8e\u4e0d\u540c\u53d6\u503c\u90fd\u662f\u76f8\u4f3c\u7684\uff0c\u6240\u4ee5\u65e0\u6cd5\u533a\u5206\u3002<\/p>\n<p>(d-f)\u662ff\u662f\u975e\u7ebf\u6027\u7684\u60c5\u51b5\uff0c\u7531\u4e0a\u56fe\u53ef\u77e5\uff0c\u5f53\u56e0\u679c\u51fd\u6570\u662f\u975e\u7ebf\u6027\u7684\u60c5\u51b5\u4e0b\uff0cP(x|y)\u4e0eP(y|x)\u7684\u503c\u5bf9\u4e8e\u4e0d\u540c\u53d6\u503c\u662f\u53ef\u4ee5\u533a\u5206\u7684\u3002\u6240\u4ee5\u57fa\u4e8e\u4e0a\u56fe\uff0c\u6211\u4eec\u6709\u7406\u7531\u8ba4\u4e3a\u56e0\u679c\u51fd\u6570\u7684\u975e\u7ebf\u6027\u6709\u52a9\u4e8e\u6211\u4eec\u8bc6\u522b\u56e0\u679c\u5173\u7cfb\u65b9\u5411\u3002<\/p>\n<p>\u8be5\u7bc7\u6587\u7ae0\u5047\u8bbe\u6240\u6709\u7684\u51fd\u6570\uff08<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"f,p_{x}(x),p_{n}(n)\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c6\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c6\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff09\u90fd\u662f\u4e09\u9636\u53ef\u5bfc\uff0c\u5728\u8fd9\u4e2a\u5047\u8bbe\u4e0b\uff0c\u6587\u7ae0\u7ed9\u51fa\u7684\u5b9a\u7406\u548c\u63a8\u8bba\u8bc1\u660e\u4e86\u5728\u4e00\u822c\u60c5\u51b5\u4e0b\u4e0d\u5b58\u5728backward model\uff08<strong>backward model\u6307\u7684\u662f\u90a3\u79cd\u56e0\u679c\u5173\u7cfb\u5728\u4e24\u4e2a\u65b9\u5411\u90fd\u6210\u7acb\u7684\u60c5\u51b5x-&gt;y\u4e0ex&lt;-y\u540c\u65f6\u5b58\u5728\uff09\u3002<\/strong><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/f1f3392299fa49fe98a71586cf168458.png\" =\"\" =\"\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c7\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c7\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<p>&nbsp;\u8fd9\u4e2a\u5b9a\u74061\u63cf\u8ff0\u7684\u5c31\u662f\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u4f1a\u5b58\u5728backward model\u3002\u5373\u5f53\u7ed9\u5b9a\u56fa\u5b9a\u7684\u51fd\u6570F\u548cV\u65f6\uff0c\u5982\u679cy\u6ee1\u8db3<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"v''(y-f(x))f'(x)\\neq 0\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c8\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c8\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff0c\u5219\u5c31\u4f1a\u5b58\u5728\u4e00\u4e2abackward model\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/99637d48252a43d7ba326ba1323109c5.png\" =\"\" =\"\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c9\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c9\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><\/p>\n<p>&nbsp;\u8fd9\u4e2a\u63a8\u8bba\u63cf\u8ff0\u7684\u5c31\u662f\u5f53\u4ec0\u4e48\u65f6\u5019\u5b58\u5728backward model\u7684\u65f6\u5019\uff0c\u6211\u4eec\u53ef\u4ee5\u53cd\u63a8\u5f97\u5230\u6570\u636e\u7684\u751f\u6210\u673a\u5236\u65f6\u7ebf\u6027\u7684\u3002<\/p>\n<p>\u4e0a\u9762\u8bb2\u5b8c\u4e86\u6a21\u578b\u7684\u53ef\u8bc6\u522b\u6027\u5982\u4f55\u5f97\u5230\u540e\uff0c\u4e0b\u9762\u5c31\u8981\u8bb2\u8bb2\u600e\u4e48\u5229\u7528\u975e\u7ebf\u6027\u53bb\u4f30\u8ba1\u6a21\u578b\u4e86\uff0c\u4e5f\u5c31\u662f\u5982\u4f55\u5f97\u5230\u56e0\u679c\u5173\u7cfb\u65b9\u5411\u4e86\uff0c\u5927\u81f4\u6b65\u9aa4\u5982\u4e0b\u3002<\/p>\n<ol>\n<li>\u9996\u5148\u9009\u53d6\u4e00\u5bf9\u6d4b\u8bd5\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u7edf\u8ba1\u72ec\u7acb\u3002<\/li>\n<li>\u5148\u5047\u8bbe\u56e0\u679c\u673a\u5236\u662f<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"y:=f(x)+n\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c10\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c10\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff0c\u7136\u540e\u6211\u4eec\u5229\u7528x\u5bf9y\u505a\u975e\u7ebf\u6027\u56de\u5f52\u5f97\u5230\u4e00\u4e2a\u62df\u5408\u51fd\u6570<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"\\tilde{f}\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c11\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c11\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff0c\u8ba1\u7b97\u76f8\u5e94\u7684\u6b8b\u5dee<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"\\hat{n} = y-\\tilde{f}(x)\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c12\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c12\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\uff0c\u901a\u8fc7\u5224\u65ad\u8fd9\u4e2a\u6b8b\u5dee\u4e0ex\u662f\u5426\u72ec\u7acb\u6765\u6d4b\u8bd5\u8fd9\u4e2a\u65b9\u5411\u662f\u5426\u4e0e\u6570\u636e\u751f\u6210\u673a\u5236\u4e00\u81f4\uff0c\u5982\u679c\u72ec\u7acb\uff0c\u5219\u53ef\u8ba4\u4e3a\u6211\u4eec\u7684\u5047\u8bbe\u662f\u5bf9\u7684\uff0c\u5982\u679c\u4e0d\u662f\uff0c\u5219\u62d2\u7edd\u8fd9\u4e2a\u5047\u8bbe\uff1b\u540c\u6837\uff0c\u6211\u4eec\u4e5f\u4f1a\u5bf9\u53e6\u4e00\u4e2a\u65b9\u5411\u505a\u5047\u8bbe\uff0c\u5373<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"x:=g(y)+n\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c13\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c13\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\u3002\u5982\u679c\u4e24\u4e2a\u65b9\u5411\u90fd\u72ec\u7acb\uff0c\u90a3\u4e48\u6211\u5c31\u9009\u53d6\u72ec\u7acb\u5206\u6570\u66f4\u9ad8\u7684\u90a3\u8fb9\u3002<\/li>\n<li>\u91cd\u590d\u4ee5\u4e0a\u6b65\u9aa4\uff0c\u76f4\u81f3\u6bcf\u5bf9\u53d8\u91cf\u90fd\u6d4b\u8bd5\u8fc7\u3002<\/li>\n<\/ol>\n<p>\u5f53\u7136\u8be5\u7b97\u6cd5\u6240\u5f97\u5230\u7684\u4f9d\u7136\u662f\u4e00\u7ec4\u7b26\u5408\u6570\u636e\u751f\u6210\u673a\u5236\u7684DAG\u3002\u4e0b\u9762\u662f\u8be5\u7b97\u6cd5\u5728\u4eff\u771f\u6570\u636e\u4e0a\u7684\u8868\u73b0\uff0c\u6211\u4eec\u7684\u6a21\u578b\u4e3a\uff1a<img decoding=\"async\" src=\"https:\/\/latex.csdn.net\/eq\" =\"\" =\"y=x+bx^3+n\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c14\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c14\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" >\u3002\u53d8\u91cfx\u4e0en\u90fd\u4ece\u9ad8\u65af\u5206\u5e03\u4e2d\u91c7\u6837\u3002\u5b9e\u9a8c\u7ed3\u679c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/7953adceb1ce462296e9403b572d89bb.png\" =\"\" =\"\" alt=\"ANM(Nonlinear causal discovery with additive noise models) (https:\/\/mushiming.com\/)  \u7b2c15\u5f20\" title=\"ANM(Nonlinear causal discovery with additive noise models)  \u7b2c15\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" 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