$\sigma$-Maximal Ancestral Graphs

TL;DR

This paper introduces $\sigma$-Maximal Ancestral Graphs, extending MAGs to represent cyclic causal relationships with latent variables, thereby broadening the scope of causal graphical models.

Contribution

It proposes $\sigma$-MAGs as a new class of graphs that can represent cyclic causal structures with latent variables, expanding the applicability of MAGs.

Findings

$\sigma$-MAGs can represent cyclic causal graphs.

Characterization of Markov equivalence classes for $\sigma$-MAGs.

Theoretical properties of $\sigma$-MAGs are established.

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.