  We provide a classification of graphical models according to their
representation as subfamilies of exponential families. Undirected graphical
models with no hidden variables are linear exponential families (LEFs),
directed acyclic graphical models and chain graphs with no hidden variables,
including Bayesian networks with several families of local distributions, are
curved exponential families (CEFs) and graphical models with hidden variables
are stratified exponential families (SEFs). An SEF is a finite union of CEFs
satisfying a frontier condition. 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 model selection is
examined.
