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Separation and Completeness Properties for Amp Chain Graph Markov Models
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 property for 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
On Separation Criterion and Recovery Algorithm for Chain Graphs
Chain graphs (CGs) give a natural unifying point of view on Markov and Bayesian networks and enlarge the potential of graphical models for description of conditional independence structures. In the paper a direct graphical separation criterion for CGs which generalizes the d-separation criterion for Bayesian networks is introduced (recalled) . It is equivalent to the classic moralization criterion for CGs and complete in the sense that for every CG there exists a probability distribution satisfying exactly independencies derivable from the CG by the separation criterion. Every class of Markov equivalent CGs can be uniquely described by a natural representative, called the largest CG. A recovery algorithm, which on basis of the (conditional) dependency model given by a CG finds the corresponding largest CG, is presented. 1 INTRODUCTION Traditional graphical models for description of probabilistic conditional independence structure use either undirected graphs (UGs), named also Markov n..