A Characterization of Markov Equivalence Classes for Directed Acyclic Graphs with Latent Variables

By Jiji Zhang. 2007

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Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly identify all arrow orientations shared by all DAGs in a Markov equivalence class, given a member of that class. For DAG models with latent variables, maximal ancestral graphs (MAGs) provide a neat representation that facilitates model search. Earlier work (Ali et al. 2005) has identified a set of orientation rules sufficient to construct all arrowheads common to a Markov equivalence class of MAGs. In this paper, we provide extra rules sufficient to construct all common tails as well. We end up with a set of orientation rules sound and complete for identifying commonalities across a Markov equivalence class of MAGs, which is particularly useful for causal inference.

Subfields: Artificial Intelligence, Statistics and Probability

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Causal inference by using invariant prediction: identification and confidence intervals (2015, background): “Such a weaker requirement is sometimes referred to as ‘modularity’ (Pearl, 2009) or what is called ‘autonomy’ (Haavelmo, 1944; Aldrich, 1989); structural equations are autonomous if, whenever we replace one of them because of an intervention, no other structural equations change; they remain invaria”