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The invariance properties of interventional distributions relative to theobservational distribution, and how these properties allow us to refine Markovequivalence classes (MECs) of DAGs, is central to causal DAG discoveryalgorithms that use both interventional and observational data. Here, we showhow the invariance properties of interventional DAG models, and thecorresponding refinement of MECs into interventional MECs, can be generalizedto mixed graphical models that allow for latent cofounders and selectionvariables. We first generalize interventional Markov equivalence to all formalindependence models associated to loopless mixed graphs. For ancestral graphs,we prove the resulting interventional MECs admit a graphical characterizationgeneralizing that of DAGs. We then define interventional distributions foracyclic directed mixed graph models, and prove that this generalization alignswith the graphical generalization of interventional Markov equivalence givenfor the formal independence models. This provides a framework for causal modeldiscovery via observational and interventional data in the presence of latentconfounders that applies even when the interventions are uncontrolled.

Distributional Invariances and Interventional Markov Equivalence for Mixed Graph Models