The Annals of Statistics

Markov equivalence for ancestral graphs

R. Ayesha Ali, Thomas S. Richardson, and Peter Spirtes

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Ancestral graphs can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under which two maximal ancestral graphs are Markov equivalent to each other, thereby extending analogous results for DAGs given by other authors. These conditions lead to an algorithm for determining Markov equivalence that runs in time that is polynomial in the number of vertices in the graph.

Article information

Ann. Statist., Volume 37, Number 5B (2009), 2808-2837.

First available in Project Euclid: 17 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68T30: Knowledge representation 05C75: Structural characterization of families of graphs
Secondary: 68T37: Reasoning under uncertainty

Directed acyclic graphs discriminating path inducing path Markov equivalence polynomial-time algorithm


Ali, R. Ayesha; Richardson, Thomas S.; Spirtes, Peter. Markov equivalence for ancestral graphs. Ann. Statist. 37 (2009), no. 5B, 2808--2837. doi:10.1214/08-AOS626.

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