🤖 AI Summary
This paper addresses the challenge of modeling high-dimensional Poisson count variables, where marginal distributions and dependence structures are inherently coupled. We propose a novel tree-structured Markov random field that—uniquely—ensures all variables have identical Poisson marginals with a common mean, while fully decoupling marginal properties from both internal dependency strength and graph topology. Leveraging probability generating functions, stochastic order theory, and expectation allocation decomposition, we derive closed-form analytical expressions for the joint distribution, the distribution of sum variables, and individual contribution measures. The model enables efficient high-dimensional sampling and establishes a complete stochastic ordering framework. Its computational complexity is linear in dimensionality, O(d). The core innovation lies in the rigorous decoupling of Poisson marginals from dependence structure, yielding a tractable, computationally scalable paradigm for statistical modeling and attribution analysis of large-scale sparse stochastic events.
📝 Abstract
A new family of tree-structured Markov random fields for a vector of discrete counting random variables is introduced. According to the characteristics of the family, the marginal distributions of the Markov random fields are all Poisson with the same mean, and are untied from the strength or structure of their built-in dependence. This key feature is uncommon for Markov random fields and most convenient for applications purposes. The specific properties of this new family confer a straightforward sampling procedure and analytic expressions for the joint probability mass function and the joint probability generating function of the vector of counting random variables, thus granting computational methods that scale well to vectors of high dimension. We study the distribution of the sum of random variables constituting a Markov random field from the proposed family, analyze a random variable's individual contribution to that sum through expected allocations, and establish stochastic orderings to assess a wide understanding of their behavior.