  We introduce a flexible and scalable class of Bayesian geostatistical models
for discrete data, based on the class of nearest neighbor mixture transition
distribution processes (NNMP), referred to as discrete NNMP. The proposed class
characterizes spatial variability by a weighted combination of first-order
conditional probability mass functions (pmfs) for each one of a given number of
neighbors. The approach supports flexible modeling for multivariate dependence
through specification of general bivariate discrete distributions that define
the conditional pmfs. Moreover, the discrete NNMP allows for construction of
models given a pre-specified family of marginal distributions that can vary in
space, facilitating covariate inclusion. In particular, we develop a modeling
and inferential framework for copula-based NNMPs that can attain flexible
dependence structures, motivating the use of bivariate copula families for
spatial processes. Compared to the traditional class of spatial generalized
linear mixed models, where spatial dependence is introduced through a
transformation of response means, our process-based modeling approach provides
both computational and inferential advantages. We illustrate the benefits with
synthetic data examples and an analysis of North American Breeding Bird Survey
data.
