Fast Bayesian Inference for Spatial Mean-Parameterized Conway–Maxwell–Poisson Models

Count data with complex features arise in many disciplines, including ecology, agriculture, criminology, medicine, and public health. Zero inflation, spatial dependence, and non-equidispersion are common features in count data. There are currently two classes of models that allow for these features—the mode-parameterized Conway–Maxwell–Poisson (COMP) distribution and the generalized Poisson model. However both require the use of either constraints on the parameter space or a parameterization that leads to challenges in interpretability. We propose spatial mean-parameterized COMP models that retain the flexibility of these models while resolving the above issues. We use a Bayesian spatial filtering approach in order to efficiently handle high-dimensional spatial data and we use reversible-jump MCMC to automatically choose the basis vectors for spatial filtering. The COMP distribution poses two additional computational challenges—an intractable normalizing function in the likelihood and no closed-form expression for the mean. We propose a fast computational approach that addresses these challenges by, respectively, introducing an efficient auxiliary variable algorithm and pre-computing key approximations for fast likelihood evaluation. We illustrate the application of our methodology to simulated and real datasets, including Texas HPV-cancer data and US vaccine refusal data. Supplementary materials for this article are available online.