Bayesian geostatistical modeling for discrete‐valued processes

We introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on nearest‐neighbor mixture processes (NNMP), referred to as discrete NNMP. To define the joint probability mass function (pmf) over a set of spatial locations, we build from local mixtures of conditional pmfs using a directed graphical model, with a directed acyclic graph that summarizes the nearest neighbor structure. The approach supports direct, flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. 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. Moreover, the framework allows for construction of models given a pre‐specified family of marginal distributions that can vary in space, facilitating covariate inclusion. 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 methodology with synthetic data examples and an analysis of North American Breeding Bird Survey data.