Nested Markov Properties for Acyclic Directed Mixed Graphs

Directed acyclic graph (DAG) models may be characterized in four different ways: via a factorization, the d-separation criterion, the moralization criterion, and the local Markov property. As pointed out by Robins [2, 1], Verma and Pearl [6], and Tian and Pearl [5], marginals of DAG models also imply equality constraints that are not conditional independences. The well-known 'Verma constraint' is an example. Constraints of this type were used for testing edges [3], and an efficient variable elimination scheme [4]. Using acyclic directed mixed graphs (ADMGs) we provide a graphical characterization of the constraints given in [5] via a nested Markov property that uses a 'fixing' transformation on graphs. We give four characterizations of our nested model that are analogous to those given for DAGs. We show that marginal distributions of DAG models obey this property.