🤖 AI Summary
This work addresses the challenge of modeling multivariate count data with exclusion or incompatibility constraints on graph-structured variables by proposing a unified graphical distribution framework. It systematically constructs, for the first time, four families of distributions—graphical multinomial, negative multinomial, hypergeometric, and negative hypergeometric—leveraging decomposable graphs to encode variable dependencies and feasible configurations. The framework supports continuous interpolation between the empty and complete graphs while preserving an explicit Markov factorization and tractable sampling schemes. Building upon graphical Dirichlet-type priors, a Bayesian hierarchical model is developed, yielding closed-form posterior and predictive distributions. The approach demonstrates both flexibility and practical utility in constrained counting tasks, such as Rydberg atom excitation experiments.
📝 Abstract
The classical multinomial, negative multinomial, hypergeometric, and negative hypergeometric distributions are naturally organized by two features of the sampling scheme: sampling with or without replacement and stopping after a fixed number of draws or a fixed number of failures. We complete the graphical analogue of this scheme for decomposable graphs by adding graphical hypergeometric and graphical negative hypergeometric distributions to the previously introduced graphical multinomial and graphical negative multinomial models in Danielewska et al. (2025). The resulting four families provide a unified parametric framework for graphical modeling of multivariate count data, in which dependence and admissible configurations are encoded by a graph. They interpolate between products of univariate distributions for the empty graph and the corresponding classical multivariate distributions for the complete graph, while retaining explicit Markov factorizations and tractable sampling representations. We further develop a unified Bayesian hierarchy based on graphical Dirichlet-type distributions, obtaining explicit posterior and predictive laws. The framework is particularly natural for count data arising under exclusion or incompatibility constraints. We discuss several such applications and illustrate its practical potential using Rydberg-atom excitation data.