  In spatial statistics, it is often assumed that the spatial field of interest
is stationary and its covariance has a simple parametric form, but these
assumptions are not appropriate in many applications. Given replicate
observations of a Gaussian spatial field, we propose nonstationary and
nonparametric Bayesian inference on the spatial dependence. Instead of
estimating the quadratic (in the number of spatial locations) entries of the
covariance matrix, the idea is to infer a near-linear number of nonzero entries
in a sparse Cholesky factor of the precision matrix. Our prior assumptions are
motivated by recent results on the exponential decay of the entries of this
Cholesky factor for Matern-type covariances under a specific ordering scheme.
Our methods are highly scalable and parallelizable. We conduct numerical
comparisons and apply our methodology to climate-model output, enabling
statistical emulation of an expensive physical model.
