Markov properties for mixed graphs

By Kayvan Sadeghi, Steffen L. Lauritzen. 2011

83 citations recorded by the source corpus.

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by $m$-separation on such graphs are compositional graphoids. We focus in particular on the subclass of ribbonless graphs which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define maximality of such graphs as well as a pairwise and a global Markov property. We prove that the global and pairwise Markov properties of a maximal ribbonless graph are equivalent for any independence model that is a compositional graphoid.

Subfields: Artificial Intelligence, Statistics and Probability, Signal Processing

View Paper: https://www.semanticscholar.org/paper/b742d2902b21f29d0207d7aad307b40f47c28698

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Citations in the corpus

Nested Markov Properties for Acyclic Directed Mixed Graphs (2012, background): “Much previous work (Ali et al., 2009; Evans and Richardson, 2010; Richardson and Spirtes, 2002; Sadeghi and Lauritzen, 2014; Wermuth, 2011; Wermuth et al., 1994) has defined Markov models for mixed graphs via independence constraints implied by hidden variable DAGs on the observed margin.”
Total positivity in Markov structures (2015, background): “port condition, in particular if it is strictly positive, and since these are suﬃcient conditions for the intersection property to hold, then pairwise and global Markov properties are equivalent; see [33]. We prove in the following result that if the pairwise-global equivalence is already establishe”