Pearl ’s well-known $d$-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces $p$-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs ( = adicyclic graphs), which include both ADGs and undirected graphs as special cases. The equivalence of p-separation to the augmentation criterion occurring in the AMP global Markov property is established, and $p$-separation is applied to prove completeness of the global Markov propertyfor AMP chain graph models. Strong completeness of the AMP Markov property is established, that is, the existence of Markov perfect distributions that satisfy those and only those conditional independences implied by the AMP property (equivalently, by $p$-separation). A linear-time algorithm for determining $p$-separation is presented.
"Separation and Completeness Properties for Amp Chain Graph Markov Models." Ann. Statist. 29 (6) 1751 - 1784, December 2001. https://doi.org/10.1214/aos/1015345961