Factorization, Inference and Parameter Learning in Discrete Amp Chain Graphs: Addendum

A probability distribution p is Markovian wrt an AMP CG G iff the following three properties hold for all C ∈ Cc(G) (Andersson et al., 2001, Theorem 2): ● C1: C⊥pNdG(C) ∖CcG(PaG(C))∣CcG(PaG(C)). ● C2: p(C ∣CcG(PaG(C))) is Markovian wrt GC . ● C3∗: For all D ⊆ C, D⊥pCcG(PaG(C)) ∖ PaG(D)∣PaG(D). Lemma 1. C1, C2 and C3∗ hold iff the following two properties hold: ● C1∗: For all D ⊆ C, D⊥pNdG(D) ∖ PaG(D)∣PaG(D). ● C2∗: p(C ∣PaG(C)) is Markovian wrt GC . Proof. First, C1∗ implies C3∗ by decomposition. Second, C1∗ implies C1 by taking D = C and applying weak union. Third, C1 and the fact that NdG(D) = NdG(C) imply D ⊥ pNdG(D) ∖ CcG(PaG(C))∣CcG(PaG(C)) by symmetry and decomposition, which together with C3∗ imply C1∗ by contraction. Finally, C2 and C2∗ are equivalent because p(C ∣PaG(C)) = p(C ∣CcG(PaG(C))) by C1∗ and decomposition.