Graphical Negative Multinomial and Multinomial Models with Dirichlet-type priors

TL;DR

This paper introduces novel discrete parametric graphical models, the graph negative multinomial and multinomial distributions, along with their conjugate Dirichlet priors, bridging a gap between continuous and discrete Bayesian graphical models.

Contribution

It proposes the first discrete parametric graphical models with Dirichlet-type priors, including their probabilistic decompositions and hyper Markov properties, extending Bayesian graphical modeling.

Findings

Derived Markov decompositions for the models

Established conjugate priors with explicit normalizing constants

Proved independence structures and hyper Markov properties

Abstract

Bayesian statistical graphical models are typically classified as 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 (the former related to the Cartier-Foata theorem for the graph genereted free quotient monoid). 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 decompositions and provide related probabilistic representations. We also introduce graphical versions of the Dirichlet and inverted Dirichlet distributions, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and establish their independence structure (a graphical version of neutrality), which yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for graphical Dirichlet laws via respective graphical versions of neutrality, which extends previously known results.