Graphical Models and Exponential Families

We show that some graphical models with no hidden variables including Bayesian networks with several families of local distributions are Curved Exponential Families (CEFs). We also show that Baysian networks with hidden variables, and several other types of graphical models including non-chordal undirected graphical models are Stratified Exponential Families (SEFs). In addition, we illustrate how one can automatically generate independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for Bayesian model selection is examined.