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December 2001 Separation and Completeness Properties for Amp Chain Graph Markov Models
Michael Levitz, David Madigan, Michael D. Perlman
Ann. Statist. 29(6): 1751-1784 (December 2001). DOI: 10.1214/aos/1015345961

Abstract

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.

Citation

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Michael Levitz. David Madigan. Michael D. Perlman. "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

Information

Published: December 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1043.62080
MathSciNet: MR1891745
Digital Object Identifier: 10.1214/aos/1015345961

Subjects:
Primary: 60K99 , 62M45
Secondary: 68R10 , 68T30

Keywords: $d$-separation , $p$-separation , acyclic directed graph , AMP model , Bayesian network , chain graph , completeness , efficient algorithm , graphical Markov model

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 6 • December 2001
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