  In spite of the diverse literature on nonstationary spatial modeling and
approximate Gaussian process (GP) methods, there are no general approaches for
conducting fully Bayesian inference for moderately sized nonstationary spatial
data sets on a personal laptop. For statisticians and data scientists who wish
to learn about spatially-referenced data and conduct posterior inference and
prediction with appropriate uncertainty quantification, the lack of such
approaches and corresponding software is a significant limitation. In this
paper, we develop methodology for implementing formal Bayesian inference for a
general class of nonstationary GPs. Our novel approach uses pre-existing
frameworks for characterizing nonstationarity in a new way that is applicable
for small to moderately sized data sets via modern GP likelihood
approximations. Posterior sampling is implemented using flexible MCMC methods,
with nonstationary posterior prediction conducted as a post-processing step. We
demonstrate our novel methods on two data sets, ranging from several hundred to
several thousand locations, and compare our methodology with related
statistical methods that provide off-the-shelf software. All of our methods are
implemented in the freely available BayesNSGP software package for R.
