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.

翻译：贝叶斯统计图模型通常被划分为连续参数型（高斯模型，其精度矩阵依赖于图结构并采用Wishart型先验）与离散非参数型（其单元格概率结构依赖于图结构并采用狄利克雷型先验）。本文通过引入两种基于有限可分解图的离散参数图模型——图负多项分布与图多项分布（前者与图生成自由商幺半群的Cartier-Foata定理相关），打破了这种二分法。这两种模型分别实现了单变量负二项分布乘积与负多项分布之间，以及二项分布乘积与多项分布之间的插值。我们推导了它们的马尔可夫分解并给出了相应的概率表示。同时引入了狄利克雷分布与逆狄利克雷分布的图模型版本，作为两类离散图马尔可夫模型的共轭先验。我们推导了两种图狄利克雷律的显式归一化常数，建立了其独立性结构（中立性的图模型版本），从而为两类贝叶斯模型提供了强超马尔可夫性质。此外，通过各自的中立性图模型版本，我们给出了图狄利克雷律的表征定理，这拓展了先前已知的结果。