The Annals of Statistics

Unifying Markov properties for graphical models

Steffen Lauritzen and Kayvan Sadeghi

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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.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2251-2278.

Received: August 2016
Revised: July 2017
First available in Project Euclid: 17 August 2018

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Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section
Secondary: 62A99: None of the above, but in this section

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


Lauritzen, Steffen; Sadeghi, Kayvan. Unifying Markov properties for graphical models. Ann. Statist. 46 (2018), no. 5, 2251--2278. doi:10.1214/17-AOS1618.

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