  The invariance properties of interventional distributions relative to the
observational distribution, and how these properties allow us to refine Markov
equivalence classes (MECs) of DAGs, is central to causal DAG discovery
algorithms that use both interventional and observational data. Here, we show
how the invariance properties of interventional DAG models, and the
corresponding refinement of MECs into interventional MECs, can be generalized
to mixed graphical models that allow for latent cofounders and selection
variables. We first generalize interventional Markov equivalence to all formal
independence models associated to loopless mixed graphs. For ancestral graphs,
we prove the resulting interventional MECs admit a graphical characterization
generalizing that of DAGs. We then define interventional distributions for
acyclic directed mixed graph models, and prove that this generalization aligns
with the graphical generalization of interventional Markov equivalence given
for the formal independence models. This provides a framework for causal model
discovery via observational and interventional data in the presence of latent
confounders that applies even when the interventions are uncontrolled.
