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.

翻译：通常，统计图模型要么是连续参数的（如由依赖图的精度矩阵参数化的高斯分布），要么是离散非参数的（具有依赖图的单元概率）。最终，这两种类型会混合使用。我们提出一种打破这种二分法的方法，通过引入两种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。这些模型分别插值于单变量负多项分布乘积与负多项分布乘积之间，以及二项分布乘积与多项分布乘积之间。我们推导了它们的马尔可夫分解，并提出了导致这两种模型的概率模型。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。我们推导了这两种图狄利克雷律的显式归一化常数，并证明其独立性结构（中性的图版本）对两个贝叶斯模型均产生强超马尔可夫性质。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特征定理。最后，我们为具有相应狄利克雷型先验的图负多项模型开发了一种贝叶斯模型选择程序。