  Bayesian statistical graphical models are typically either continuous and
parametric (Gaussian, parameterized by the graph-dependent precision matrix
with Wishart-type priors) or discrete and non-parametric (with graph-dependent
structure of probabilities of cells and Dirichlet-type priors). We propose to
break this dichotomy by introducing two discrete parametric graphical models on
finite decomposable graphs: the graph negative multinomial and the graph
multinomial distributions. These models interpolate between the product of
univariate negative binomial laws and the negative multinomial distribution,
and between the product of binomial laws and the multinomial distribution,
respectively. We derive their Markov decomposition and present related
probabilistic models representations. We also introduce graphical versions of
the Dirichlet distribution and inverted Dirichlet distribution, which serve as
conjugate priors for the two discrete graphical Markov models. We derive
explicit normalizing constants for both graphical Dirichlet laws and
demonstrate that their independence structure (a graphical version of
neutrality) yields a strong hyper Markov property for both Bayesian models. We
also provide characterization theorems for graphical Dirichlet laws via strong
hyper Markov property. Finally, we develop a model selection procedure for the
Bayesian graphical negative multinomial model with respective Dirichlet-type
priors.
