$σ$-Maximal Ancestral Graphs

Binghua Yao, Joris Marten Mooij
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:4775-4805, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-yao25a, title = {$σ$-Maximal Ancestral Graphs}, author = {Yao, Binghua and Mooij, Joris Marten}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {4775--4805}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/yao25a/yao25a.pdf}, url = {https://proceedings.mlr.press/v286/yao25a.html}, 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.} }
Endnote
%0 Conference Paper %T $σ$-Maximal Ancestral Graphs %A Binghua Yao %A Joris Marten Mooij %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-yao25a %I PMLR %P 4775--4805 %U https://proceedings.mlr.press/v286/yao25a.html %V 286 %X 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.
APA
Yao, B. & Mooij, J.M.. (2025). $σ$-Maximal Ancestral Graphs. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:4775-4805 Available from https://proceedings.mlr.press/v286/yao25a.html.

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