Discrete Exponential Bayesian Networks: An Extension of Bayesian Networks to Discrete Natural Exponential Families

In this paper, we develop the notion of discrete exponential Bayesian network, parametrization of Bayesian networks (BNs) using more general discrete quadratic exponential families instead of usual multinomial ones. We then introduce a family of prior distributions which generalizes the Dirichlet prior classically used with discrete Bayesian network. We develop the posterior distribution for our discrete exponential BNs leading to bayesian estimations of the parameters of our models and one new scoring function extending the Bayesian Dirichlet score used for structure learning. These theoretical results are finally illustrated for Poisson and Negative Binomial BNs.