2023/01/15 by Danielewska, Iza, Bartosz Kołodziejek, Kołodziejek, Bartosz +4
Computer Science · Mathematics · #62E10 (Secondary) #62H22 (Primary) 60E05 #Bayesian Methods and Mixture Models #Bayesian Modeling and Causal Inference #FOS: Mathematics #Probability (math.PR) #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2301.06058
openalex publication_date 2023/01/15 · openalex created_date 2023/01/19 · openalex updated_date 2026/08/03
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.