  Current approaches for modeling discrete-valued outcomes associated with
spatially-dependent areal units incur computational and theoretical challenges,
especially in the Bayesian setting when full posterior inference is desired. As
an alternative, we propose a novel statistical modeling framework for this data
setting, namely a mixture of directed graphical models (MDGMs). The components
of the mixture, directed graphical models, can be represented by directed
acyclic graphs (DAGs) and are computationally quick to evaluate. The DAGs
representing the mixture components are selected to correspond to an undirected
graphical representation of an assumed spatial contiguity/dependence structure
of the areal units, which underlies the specification of traditional modeling
approaches for discrete spatial processes such as Markov random fields (MRFs).
We introduce the concept of compatibility to show how an undirected graph can
be used as a template for the structural dependencies between areal units to
create sets of DAGs which, as a collection, preserve the structural
dependencies represented in the template undirected graph. We then introduce
three classes of compatible DAGs and corresponding algorithms for fitting MDGMs
based on these classes. In addition, we compare MDGMs to MRFs and a popular
Bayesian MRF model approximation used in high-dimensional settings in a series
of simulations and an analysis of ecometrics data collected as part of the
Adolescent Health and Development in Context Study.
