  Directed acyclic graph (DAG) models, also called Bayesian networks, impose
conditional independence constraints on a multivariate probability
distribution, and are widely used in probabilistic reasoning, machine learning
and causal inference. If latent variables are included in such a model, then
the set of possible marginal distributions over the remaining (observed)
variables is generally complex, and not represented by any DAG. Larger classes
of mixed graphical models, which use multiple edge types, have been introduced
to overcome this; however, these classes do not represent all the models which
can arise as margins of DAGs. In this paper we show that this is because
ordinary mixed graphs are fundamentally insufficiently rich to capture the
variety of marginal models.
  We introduce a new class of hyper-graphs, called mDAGs, and a latent
projection operation to obtain an mDAG from the margin of a DAG. We show that
each distinct marginal of a DAG model is represented by at least one mDAG, and
provide graphical results towards characterizing when two such marginal models
are the same. Finally we show that mDAGs correctly capture the marginal
structure of causally-interpreted DAGs under interventions on the observed
variables.
