Appendix A Removable Variables In this section, we first prove the proposed graphical representation for a removable variable in a MAG

(Theorem 1). A.1 Graphical representation Theorem 1. V ertex X is removable in a MAG M over the variables V, if and only if 1. for any Y 2 Adj ( X) and Z 2 Ch ( X) [ N ( X) \{ Y }, Y and Z are adjacent, and 2. Let H denote the induced subgraph of M over V \{ X } . Since X is removable in M, by definition of removability, ( Y? M, Lemma 6 implies that u is not m-connecting relative to W in H . (: Lemma 6 implies that u is not m-connecting relative to W in M . This contradiction proves that X cannot have a descendant in { Y,Z }[ W, which implies that X blocks u in M .