  Hierarchical spatial models are very flexible and popular for a vast array of
applications in areas such as ecology, social science, public health, and
atmospheric science. It is common to carry out Bayesian inference for these
models via Markov chain Monte Carlo (MCMC). Each iteration of the MCMC
algorithm is computationally expensive due to costly matrix operations. In
addition, the MCMC algorithm needs to be run for more iterations because the
strong cross-correlations among the spatial latent variables result in slow
mixing Markov chains. To address these computational challenges, we propose a
projection-based intrinsic conditional autoregression (PICAR) approach, which
is a discretized and dimension-reduced representation of the underlying spatial
random field using empirical basis functions on a triangular mesh. Our approach
exhibits fast mixing as well as a considerable reduction in computational cost
per iteration. PICAR is computationally efficient and scales well to high
dimensions. It is also automated and easy to implement for a wide array of
user-specified hierarchical spatial models. We show, via simulation studies,
that our approach performs well in terms of parameter inference and prediction.
We provide several examples to illustrate the applicability of our method,
including (i) a high-dimensional cloud cover dataset that showcases its
computational efficiency, (ii) a spatially varying coefficient model that
demonstrates the ease of implementation of PICAR in the probabilistic
programming languages stan and nimble, and (iii) a watershed survey example
that illustrates how PICAR applies to models that are not amenable to efficient
inference via existing methods.
