  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 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 a spatial mean-parameterized COMP model that
retains 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.

Methodology;Applications;Computation