{"id":2780,"date":"2021-12-22T15:43:53","date_gmt":"2021-12-22T07:43:53","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=2780"},"modified":"2022-03-02T16:51:26","modified_gmt":"2022-03-02T08:51:26","slug":"causal-inference-in-statistics-by-judea-pearl","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=2780","title":{"rendered":"Causal Inference in Statistics"},"content":{"rendered":"\n<p>Base: Correlation does not mean casualty.<\/p>\n\n\n\n<ul><li>If X and Y are statistically dependent, X does not necessarily cause Y (or Y cause X). \u76f8\u5173\u6027\u4e0d\u4ee3\u8868\u6709\u56e0\u679c\u6027<\/li><li>If X causes Y, then X &amp; Y are very likely to be statistically dependent (but not always, there is extreme condition). \u4f46\u662f\u56e0\u679c\u6027\u4ee3\u8868\u76f8\u5173\u6027<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Study 1. V Structure:<\/h4>\n\n\n\n<ul><li>Chain<\/li><\/ul>\n\n\n\n<p>$$ X\\rightarrow Y\\rightarrow Z $$<\/p>\n\n\n\n<p><strong>Z and X are likely dependent. However, Z and X are independent, conditional on Y.<\/strong><\/p>\n\n\n\n<p>$$ P(Z=z|X=x,Y=c)=P(Z=z|Y=c) $$i.e.<\/p>\n\n\n\n<p>i.e.<\/p>\n\n\n\n<p>\\(f_x: X=u_x\\)<\/p>\n\n\n\n<p>\\(f_y: Y=84-X+u_Y := c\\)<\/p>\n\n\n\n<p>\\(f_z: Z=100\\underbrace{Y}_{c}+u_z\\)<\/p>\n\n\n\n<p>Now, Z and X are independent.<\/p>\n\n\n\n<p>Therefore, we know, in the Chain:<\/p>\n\n\n\n<p>$$ X\\equiv Z$$<\/p>\n\n\n\n<p>$$ X\\bot Z|Y $$<\/p>\n\n\n\n<ul><li>Folk<\/li><\/ul>\n\n\n\n<p>$$ Y\\leftarrow X\\rightarrow Z $$<\/p>\n\n\n\n<p><strong>Y and Z are likely dependent. However, Y and Z are independent conditional on X.<\/strong><\/p>\n\n\n\n<p>$$P(Z=z|Y=y, X=c)=P(Z=z|X=c)$$<\/p>\n\n\n\n<p>While conditioning on intermediate node X, then Z and Y are independent.<\/p>\n\n\n\n<p>$$Y\\bot Z|X$$<\/p>\n\n\n\n<ul><li>Collider<\/li><\/ul>\n\n\n\n<p>$$ X\\rightarrow Z\\leftarrow Y $$<\/p>\n\n\n\n<p>X and Y are independent. However, X and Y are dependent conditional on Z.<\/p>\n\n\n\n<p>$$ P(X=x|Y=y, Z=c)\\neq P(X=x|Z=c) $$<\/p>\n\n\n\n<p>i.e.<\/p>\n\n\n\n<p>If we know \\(Z=X+Y+u_Z:=c\\), then \\( X=c-Y-u_Z\\), and thus X and Y become dependent conditional on \\(Z=c\\). Otherwise, \\(X=u_X\\) and \\(Y=u_Y\\).<\/p>\n\n\n\n<p>Once, conditioning on \\(Z\\), the way gets connected. Otherwise (unconditional), we get independent.<\/p>\n\n\n\n<p>P.S. <strong>Descendent of Z:<\/strong><\/p>\n\n\n\n<p>$$ X (or\\ Y)\\rightarrow Z\\rightarrow W $$ <\/p>\n\n\n\n<p>Similarly, we get in the Collider:<\/p>\n\n\n\n<p>$$ X\\bot Y  $$<\/p>\n\n\n\n<p>$$ X\\equiv Y|Z $$<\/p>\n\n\n\n<p>$$ X\\equiv Y |W $$<\/p>\n\n\n\n<ul><li>See notes for further studies. <\/li><\/ul>\n\n\n\n<div data-wp-interactive=\"\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!selectors.core.file.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/12\/Casual-Inference.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Embed of Casual-Inference..\"><\/object><a href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/12\/Casual-Inference.pdf\">Casual-Inference<\/a><a href=\"https:\/\/fanyuzhao.com\/wp-content\/uploads\/2021\/12\/Casual-Inference.pdf\" class=\"wp-block-file__button\" download>Download<\/a><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Reference<\/h4>\n\n\n\n<p>Pearl, J., Glymour, M. and Jewell, N.P., 2016.\u00a0<em>Causal inference in statistics: A primer<\/em>. John Wiley &amp; Sons.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Base: Correlation does not mean casualty. If X and Y are statistically dependent, X does not necessarily cause Y (or Y cause X). \u76f8\u5173\u6027\u4e0d\u4ee3\u8868\u6709\u56e0\u679c\u6027 If X causes Y, then X &amp; Y are very likely to be statistically dependent (but not always, there is extreme condition). \u4f46\u662f\u56e0\u679c\u6027\u4ee3\u8868\u76f8\u5173\u6027 Study 1. V Structure: Chain $$ X\\rightarrow Y\\rightarrow &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=2780\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Causal Inference in Statistics<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2780"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2780"}],"version-history":[{"count":57,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2780\/revisions"}],"predecessor-version":[{"id":2838,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2780\/revisions\/2838"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2780"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2780"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}