Tree-structured Markov random fields with Poisson marginal distributions

Having graphs underlie multivariate distributions performs a translation of their vast range of topologies to a rich variety of dependence schemes; this is the premise of probabilistic graphical models. The graph, through its edges and its vertices, serves as a representation of the dependence relations knitting the random variables to one another. Among others, [Koller and Friedman, 2009] and [Maathuis et al., 2018] dive deeply into probabilistic graphical models, with much emphasis on Bayesian networks and Markov random fields (MRFs), also called Markov networks or undirected graphical models. In [Besag, 1974], a seminal paper on MRFs, the author defines various families through their conditional distributions: the distribution of a random variable is specified given the value taken by its neighbours according to the underlying graph. A family that has drawn particular attention to represent a vector of count random variables is Besag's auto-Poisson MRFs, also called Poisson graphical models, where each vertex's conditional distribution is Poisson, the values taken by the neighbouring vertices' random variables influencing its mean parameter.