$σ$-Maximal Ancestral Graphs
This work addresses a foundational gap in causal discovery for researchers dealing with cyclic relationships, though it appears incremental as it extends existing MAG theory.
The paper tackles the limitation of Maximal Ancestral Graphs (MAGs) in excluding cyclic causal relationships by introducing σ-Maximal Ancestral Graphs (σ-MAGs), which provide an abstract representation for possibly cyclic directed graphs with latent variables and characterize their Markov equivalence classes.
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 ''$σ$-Maximal Ancestral Graphs'' (''$σ$-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.