$σ$-Maximal Ancestral Graphs

2025-06-30

AI Summary

Traditional maximal ancestral graphs (MAGs) are restricted to acyclic causal models with latent variables and cannot represent cyclic causal dependencies. Method: We introduce σ-MAGs—the first generalization of MAGs that accommodates both directed cycles and latent variables. By defining σ-separation, a generalized notion of ancestry, and a maximality condition, we establish a rigorous Markov semantics and characterize the corresponding Markov equivalence class. Contribution/Results: We prove that σ-MAGs faithfully encode the conditional independence structure of cyclic directed graphs—even in the presence of latent confounders—and provide a sound, complete, and minimal representation for causal discovery under cyclic dependencies and unobserved variables. This work overcomes the fundamental acyclicity constraint inherent in classical MAGs, enabling equivalence-class–based modeling and inference for cyclic causal systems for the first time.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 ''$σ$-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.

Problem

• Extends MAGs to represent cyclic causal relationships
• Introduces σ-MAGs for abstract representation of Directed Graphs
• Characterizes Markov equivalence classes for σ-MAGs

Innovation

• Introduces σ-MAGs for cyclic causal relationships
• Extends MAGs to represent Directed Graphs
• Characterizes Markov equivalence classes