Interventional Markov Equivalence for Mixed Graph Models

We study the problem of characterizing Markov equivalence of graphical models under general interventions. For DAGs, this problem is solved using data from an interventional setting to refine MECs of DAGs into smaller, interventional MECs. A recent graphical characterization of interventional MECs of DAGs relates to their global Markov property. Motivated by this, we generalize interventional MECs to all loopless mixed graphs via their global Markov property and generalize the graphical characterization given for DAGs to ancestral graphs. We also extend the notion of interventional Markov equivalence probabilistically: via invariance properties of distributions Markov to acyclic directed mixed graphs (ADMGs). We show that this generalization aligns with the standard causal interpretation of ADMGs. Finally, we show the two generalizations coincide at their intersection, thereby completely generalizing the characterization for DAGs to directed ancestral graphs.