## Separation and completeness properties for AMP chain graph Markov models.(English)Zbl 1043.62080

Summary: J. Pearl’s [Probabilistic reasoning in intelligent systems: networks of plausible inference. (1989; Zbl 0746.68089)] 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 S. A. Andersson, D. Madigan and M. D. Perlman (AMP) alternative Markov property for chain graphs (= adicyclic graphs) [Scand. J. Stat. 24, 81–102 (1997; Zbl 0918.60050)], 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.

### MSC:

 62M45 Neural nets and related approaches to inference from stochastic processes 68R10 Graph theory (including graph drawing) in computer science 05C90 Applications of graph theory 65C60 Computational problems in statistics (MSC2010)

### Keywords:

Bayesian networks; completeness; efficient algorithm

### Citations:

Zbl 0746.68089; Zbl 0918.60050