k l PAl,k,thetotalcausaleffectfromitoj

Suppose variables follow random measurement error model defined inEquation(1). W S. Since observed variables are all leaf nodes of their respective latent nodes, for every undirected pathplinkingXi andXj,pmust beinform ofXi Xi Xj Xj. Hence,W must be in latent nodes, and onlycase 1)is possible, which means that there exists a collider on everyp linking Xi and Xj, and thusXi and Xj is also d-separated by (conclusion 1). Specifically on case1),W Sisobvious(sinceW XandS X).And,bydes(W) S=,wehaveW S and des(W) S = (easy to show sinceS des( S)), and thus among latent nodes, there is Xi d Xj| S(conclusion2). Based on Theorem 6, we can readily give proof to Theorem 2. Note that in our setting where X=AX+E=BEwithB=(I A) 1,weknowthatAj,iisthee(i,j)above,andBisexactly M,withBj,i = Interestingly, we find that our defined vertex cut has connection with trek-separation [47], i.e. "S isavertexcut fromAnc(Z)toY"isequivalent to"(,S)t-separates(Z,Y)"(see Appendix C).