$\sigma$-Maximal Ancestral Graphs

Binghua Yao, Joris Mooij

Keywords: Causal Discovery, Graphical Models, Maximal Ancestral Graphs (MAGs), Markov Equivalence, m-separation, $\sigma$-separation

TL;DR: We introduce $\sigma$-Maximal Ancestral Graphs ($\sigma$-MAGs), a new class of graphical models that extend MAGs to represent possibly cyclic directed graphs with latent (selection) variables, and we characterize their Markov equivalence classes.

Abstract: Maximal Ancestral Graphs (MAGs) provide an abstract representation of Directed Acyclic Graphs (DAGs) with latent (selection) variables. These graphical objects encode information about ancestral relations and d-separations of the DAGs they represent. This abstract representation has been used amongst others to prove the soundness and completeness of the FCI algorithm for causal discovery, and to derive a do-calculus for its output. One significant inherent limitation of MAGs is that they rule out the possibility of cyclic causal relationships. In this work, we address that limitation. We introduce and study a class of graphical objects that we coin "$\sigma$-Maximal Ancestral Graphs" ("$\sigma$-MAGs"). We show how these graphs provide an abstract representation of (possibly cyclic) Directed Graphs (DGs) with latent (selection) variables, analogously to how MAGs represent DAGs. We study the properties of these objects and provide a characterization of their Markov equivalence classes.

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Submission Number: 366