Discrete parametric graphical models with Dirichlet type priors

Typically, statistical graphical models are either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix) or discrete and non-parametric (with graph-dependent probabilities of cells). Eventually, the two types are mixed. We propose a way 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 multinomial and negative multinomial distributions, and between the product of binomial and multinomial distributions, respectively. We derive their Markov decomposition and present probabilistic models leading to both. Additionally, we 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 the generalized Dirichlet distributions via strong hyper Markov property. Finally, we develop a Bayesian model selection procedure for the graphical negative multinomial model with respective Dirichlet-type priors.