Open Access
August 1998 On chain graph models for description of conditional independence structures
Remco R. Bouckaert, Milan Studený
Ann. Statist. 26(4): 1434-1495 (August 1998). DOI: 10.1214/aos/1024691250

Abstract

A chain graph (CG) is a graph admitting both directed and undirected edges with (partially) directed cycles forbidden. It generalizes both the concept of undirected graph (UG) and the concept of directed acyclic graph (DAG). A chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid, that is, in the form of a list of statements “$X$ is independent of $Y$ given $Z$” obeying a set of five properties (axioms). An input list of independency statements for every CG is defined and it is shown that the classic moralization criterion for CGs embraces exactly the graphoid closure of the input list. A new direct separation criterion for reading independency statements from a CG is introduced and shown to be equivalent to the moralization criterion. Using this new criterion, it is proved that for every CG, there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph. Thus, both criteria are shown to be complete and the use of CGs as tools for description of conditional independence structures is justified.

Citation

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Remco R. Bouckaert. Milan Studený. "On chain graph models for description of conditional independence structures." Ann. Statist. 26 (4) 1434 - 1495, August 1998. https://doi.org/10.1214/aos/1024691250

Information

Published: August 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0930.62066
MathSciNet: MR1647685
Digital Object Identifier: 10.1214/aos/1024691250

Subjects:
Primary: 62H99
Secondary: 62H05 , 68R10

Keywords: chain graph , completeness , Conditional independence , c-separation , Markovian distribution

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 4 • August 1998
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