Appendix ARemovable Variables

In this section, we first prove the proposed graphical representation for a removable variable in a MAGM (Theorem 1). Then, we discuss how this representation reduces to Theorem 5 of [11] in the case of DAGs. Throughout our proofs, we say a path between X and Y is blocked by a set Wif it is not m-connecting relative to W. In this case, there exists a non-collider W on the path which is a member of W, or there exists a collider W on the path such that W/2 Anc({X,Y }[ W). In both cases we say W blocks this path with respect to W, or W blocks the path in short when W is clear from the context. We say X is a descendant of Y if Y 2Anc(X), and we denote by DeM(X) the set of descendants of X in the MAGM, and De(X) whenever the graph is clear from the context. A.1 Graphical representation Theorem 1. Vertex X is removable in a MAGM over the variables V, if and only if 1. for any Y 2Adj(X) and Z 2Ch(X)[N(X)\{Y}, Y and Z are adjacent, and 2. for any collider path u =( X,V1,...,V m,Y) and Z 2 V\{X,Y,V1,...,V m} such that {X,V1,...,V m} Pa(Z), Y and Z are adjacent. Let H denote the induced subgraph of M over V\{X}. For any W V\{X,Y,Z}, (Z,X,Y) is an m-connecting path relative to W in M, as X is a non-collider and X/2W. That is, no such W can m-separate Y and Z. Since X is removable in M, by definition of removability, (Y?Z|W)M ()(Y?Z|W)H. Again for any W V\{X,Y,Z}, (Z,X,V1,...,V m,Y) is an m-connecting path relative to W in M since I) every collider on this path is a parent (and therefore an ancestor) of Z, and II) X/2W and X is the only non-collider on this path. That is, no such W can m-separate Y and Z. Since X is removable in M, Equation 8 implies that Y and Z have no m-separating sets in H. Hence, Y is adjacent to Z in H, and therefore, in M. if part: We need to prove that for any Y,Z 2V\{X} and any W V\{X,Y,Z}, (Y?Z|W)M ()(Y?Z|W)H.