{"id":48,"date":"2008-05-04T18:00:11","date_gmt":"2008-05-05T01:00:11","guid":{"rendered":"http:\/\/www.mii.ucla.edu\/causality\/?p=58"},"modified":"2008-05-04T18:00:11","modified_gmt":"2008-05-05T01:00:11","slug":"alternative-proof-of-the-back-door-criterion","status":"publish","type":"post","link":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/2008\/05\/04\/alternative-proof-of-the-back-door-criterion\/","title":{"rendered":"Alternative Proof of the Back-Door Criterion"},"content":{"rendered":"<p>Consider a Markovian model [tex]$G$[\/tex] in which [tex]$T$[\/tex] stands for the set of parents of [tex]$X$[\/tex]. &nbsp;From [tex]{em Causality}[\/tex], Eq.~(3.13), we know that the causal effect of [tex]$X$[\/tex] on [tex]$Y$[\/tex] is given by<\/p>\n<p>[tex]begin{equation} P(y|hat{x}) = sum_{t in T} P(y|x,t) P(t) %% eq 1 &nbsp;label{ch11-eq-a} end{equation}[\/tex] (1).<\/p>\n<p>Now assume some members of [tex]$T$[\/tex] are unobserved, and we seek another set [tex]$Z$[\/tex] of observed variables, to replace [tex]$T$[\/tex] so that<\/p>\n<p>[tex]begin{equation} P(y|hat{x}) = sum_{z in Z} P(y|x,Z) P(z) %% eq 2 &nbsp;label{ch11-eq-b} end{equation}[\/tex] (2).<\/p>\n<p>It is easily verified that (2) follow from (1) if [tex]$Z$[\/tex] satisfies:<\/p>\n<p>&nbsp;<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/causality.cs.ucla.edu\/blog\/latexrenderer\/pictures\/eq1.png\" alt=\" \" width=\"159\" height=\"48\" \/>.<\/p>\n<p>Indeed, conditioning on [tex]$Z$[\/tex], ([tex]$i$[\/tex]) permits us to rewrite (1) as [tex][ P(y|hat{x}) = sum_{t} P(t) sum_z P(y|z,x) P(z|t,x) ][\/tex] and ([tex]$ii$[\/tex]) further yields [tex]$P(z|t,x)=P(z|t)$[\/tex] from which (2) follows. It is now a purely graphical exercize to prove that the back-door criterion implies ([tex]$i$[\/tex]) and ([tex]$ii$[\/tex]). Indeed, ([tex]$ii$[\/tex]) follows directly from the fact that [tex]$Z$[\/tex] consists of nondescendants of [tex]$X$[\/tex], while the blockage of all back-door path by [tex]$Z$[\/tex] implies <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/causality.cs.ucla.edu\/blog\/latexrenderer\/pictures\/eq2.png\" alt=\" \" width=\"120\" height=\"21\" \/>, hence ([tex]$i$[\/tex]). This follows from observing that any path from [tex]$Y$[\/tex] to [tex]$T$[\/tex] in [tex]$G$[\/tex] that is unblocked by [tex]${X,Z}$[\/tex] can be extended to a back-door path from [tex]$Y$[\/tex] to [tex]$X$[\/tex], unblocked by [tex]$Z$[\/tex].<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a Markovian model [tex]$G$[\/tex] in which [tex]$T$[\/tex] stands for the set of parents of [tex]$X$[\/tex]. &nbsp;From [tex]{em Causality}[\/tex], Eq.~(3.13), we know that the causal effect of [tex]$X$[\/tex] on [tex]$Y$[\/tex] is given by [tex]begin{equation} P(y|hat{x}) = sum_{t in T} P(y|x,t) P(t) %% eq 1 &nbsp;label{ch11-eq-a} end{equation}[\/tex] (1). Now assume some members of [tex]$T$[\/tex] are unobserved, [&hellip;]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-48","post","type-post","status-publish","format-standard","hentry","category-back-door-criterion"],"_links":{"self":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/48","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=48"}],"version-history":[{"count":0,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/48\/revisions"}],"wp:attachment":[{"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=48"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=48"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/causality.cs.ucla.edu\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=48"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}