  We propose a directed acyclic hypergraph framework for a probabilistic
graphical model that we call Bayesian hypergraphs. The space of directed
acyclic hypergraphs is much larger than the space of chain graphs. Hence
Bayesian hypergraphs can model much finer factorizations than Bayesian networks
or LWF chain graphs and provide simpler and more computationally efficient
procedures for factorizations and interventions. Bayesian hypergraphs also
allow a modeler to represent causal patterns of interaction such as Noisy-OR
graphically (without additional annotations). We introduce global, local and
pairwise Markov properties of Bayesian hypergraphs and prove under which
conditions they are equivalent. We define a projection operator, called shadow,
that maps Bayesian hypergraphs to chain graphs, and show that the Markov
properties of a Bayesian hypergraph are equivalent to those of its
corresponding chain graph. We extend the causal interpretation of LWF chain
graphs to Bayesian hypergraphs and provide corresponding formulas and a
graphical criterion for intervention.
