We present 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. 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, analyze their independence structure, and demonstrate that this implies strong hyper Markov property for the Bayesian models. We also provide characterization theorems for the generalized Dirichlet distributions via strong hyper Markov property. Finally we apply our findings to develop a Bayesian model selection procedure for the graphical negative multinomial model with respective Dirichlet-type priors.

翻译：本文提出了两类定义在有限可分解图上的离散参数图模型：图负多项分布和图多项分布。这些模型分别在内插单变量负多项分布与负多项分布乘积、以及二项分布与多项分布乘积之间建立联系。我们推导了其马尔可夫分解，并给出了产生这两类分布的概率模型。此外，我们引入了狄利克雷分布和逆狄利克雷分布的图形式版本，它们作为这两个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分布的显式归一化常数，分析了其独立性结构，并证明这在贝叶斯模型中蕴含强超马尔可夫性质。我们还通过强超马尔可夫性质给出了广义狄利克雷分布的表征定理。最后，我们将研究成果应用于开发适用于带相应狄利克雷型先验的图负多项模型的贝叶斯模型选择程序。