Markov Equivalence Classes for Maximal Ancestral Graphs
نویسندگان
چکیده
Ancestral graphs provide a class of graphs that can encode conditional independence re lations that arise in directed acyclic graph (DAG) models with latent and selection vari ables, corresponding to marginalization and conditioning. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We introduce a simple representation of a Markov equiv alence class of ancestral graphs, thereby fa cilitating the model search process for some given data. More specifically, we define a join operation on ancestral graphs which will as sociate a unique graph with an equivalence class. We also extend the separation crite rion for ancestral graphs (which is an exten sion of d-separation) and provide a proof of the pairwise Markov property for joined an cestral graphs. Proving the pairwise Markov property is the first step towards developing a global Markov property for these graphs. The ultimate goal of this work is to ob tain a full characterization of the structure of Markov equivalence classes for maximal an cestral graphs, thereby extending analogous results for DAGs given by Frydenberg (1990), Verma and Pearl (1991), Chickering (1995) and Andersson et a!. ( 1997).
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