A Characterization of Markov Equivalence Classes for Directed Acyclic Graphs with Latent Variables
This work is incremental, as it extends earlier rules for arrowheads to include tails, improving the representation of causal models with latent variables.
The paper tackles the problem of characterizing Markov equivalence classes for directed acyclic graphs with latent variables by providing a complete set of orientation rules to identify all common arrowheads and tails, which is useful for causal inference.
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.