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.

翻译：贝叶斯统计图模型通常分为两类：连续参数化模型（高斯模型，由依赖于图的精度矩阵参数化，并采用Wishart型先验）与离散非参数化模型（具有依赖于图的单元格概率结构及狄利克雷型先验）。本文通过引入两类定义在有限可分解图上的离散参数化图模型——图负多项分布与图多项分布，打破了这一二分法。这两类模型分别实现了单变量负二项分布的乘积与负多项分布之间的插值，以及二项分布的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性质，并给出了相关的概率模型表示。本文还引入了狄利克雷分布与逆狄利克雷分布的图版本，作为这两类离散图马尔可夫模型的共轭先验。我们推导了这两类狄利克雷图律的显式归一化常数，并证明其独立结构（中性的图版本）赋予两个贝叶斯模型强超马尔可夫性质。同时，通过强超马尔可夫性质给出了狄利克雷图律的特征定理。最后，我们针对具有相应狄利克雷型先验的贝叶斯图负多项模型，开发了一套模型选择流程。