  Acyclic directed mixed graphs (ADMGs) are graphs that contain directed
($\rightarrow$) and bidirected ($\leftrightarrow$) edges, subject to the
constraint that there are no cycles of directed edges. Such graphs may be used
to represent the conditional independence structure induced by a DAG model
containing hidden variables on its observed margin. The Markovian model
associated with an ADMG is simply the set of distributions obeying the global
Markov property, given via a simple path criterion (m-separation). We first
present a factorization criterion characterizing the Markovian model that
generalizes the well-known recursive factorization for DAGs. For the case of
finite discrete random variables, we also provide a parameterization of the
model in terms of simple conditional probabilities, and characterize its
variation dependence. We show that the induced models are smooth. Consequently,
Markovian ADMG models for discrete variables are curved exponential families of
distributions.
