Open Access
October 2018 Unifying Markov properties for graphical models
Steffen Lauritzen, Kayvan Sadeghi
Ann. Statist. 46(5): 2251-2278 (October 2018). DOI: 10.1214/17-AOS1618


Several types of graphs with different conditional independence interpretations—also known as Markov properties—have been proposed and used in graphical models. In this paper, we unify these Markov properties by introducing a class of graphs with four types of edges—lines, arrows, arcs and dotted lines—and a single separation criterion. We show that independence structures defined by this class specialize to each of the previously defined cases, when suitable subclasses of graphs are considered. In addition, we define a pairwise Markov property for the subclass of chain mixed graphs, which includes chain graphs with the LWF interpretation, as well as summary graphs (and consequently ancestral graphs). We prove the equivalence of this pairwise Markov property to the global Markov property for compositional graphoid independence models.


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Steffen Lauritzen. Kayvan Sadeghi. "Unifying Markov properties for graphical models." Ann. Statist. 46 (5) 2251 - 2278, October 2018.


Received: 1 August 2016; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964332
MathSciNet: MR3845017
Digital Object Identifier: 10.1214/17-AOS1618

Primary: 62H99
Secondary: 62A99

Keywords: $c$-separation , $d$-separation , $m$-separation , AMP Markov property , chain graph , compositional graphoid , independence model , LWF Markov property , mixed graph , pairwise Markov property , regression chain Markov property

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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