Bayesian Conway–Maxwell–Poisson regression models for overdispersed and underdispersed counts

Bayesian models that can handle both overdispersed and underdispersed counts are rare in the literature, perhaps because full probability distributions for dispersed counts are rather difficult to construct. This note takes a first look at Bayesian Conway–Maxwell–Poisson regression models that can handle both overdispersion and underdispersion yet retain the parsimony and interpretability of classical count models. The focus is on providing an explicit demonstration of Bayesian regression inferences for dispersed counts via a Metropolis–Hastings algorithm. We illustrate the approach on two data analysis examples and demonstrate some favorable frequentist properties via a simulation study.