# σ-Maximal Ancestral Graphs

arXiv: https://arxiv.org/abs/2507.00093 · HTML: https://mlsanity.com/paper/2507.00093/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 28 references that are in the map.
- First posted on arXiv 30 Jun 2025.

## Builds on (most cited first, 8 of 8)

- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Constraint-Based Causal Discovery using Partial Ancestral Graphs in the presence of Cycles](https://mlsanity.com/paper/2005.00610/) (2020)
- [Directed Cyclic Graphical Representations of Feedback Models](https://mlsanity.com/paper/1302.4982/) (2013)
- [A Constraint-Based Algorithm For Causal Discovery with Cycles, Latent Variables and Selection Bias](https://mlsanity.com/paper/1805.02087/) (2018)
- [Establishing Markov Equivalence in Cyclic Directed Graphs](https://mlsanity.com/paper/2309.03092/) (2023)
- [A Constraint Propagation Approach to Probabilistic Reasoning](https://mlsanity.com/paper/1304.3422/) (2013)

## Abstract (arXiv)

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.