  Gaussian Markov random fields (GMRFs) are popular for modeling dependence in
large areal datasets due to their ease of interpretation and computational
convenience afforded by the sparse precision matrices needed for random
variable generation. Typically in Bayesian computation, GMRFs are updated
jointly in a block Gibbs sampler or componentwise in a single-site sampler via
the full conditional distributions. The former approach can speed convergence
by updating correlated variables all at once, while the latter avoids solving
large matrices. We consider a sampling approach in which the underlying graph
can be cut so that conditionally independent sites are updated simultaneously.
This algorithm allows a practitioner to parallelize updates of subsets of
locations or to take advantage of `vectorized' calculations in a high-level
language such as R. Through both simulated and real data, we demonstrate
computational savings that can be achieved versus both single-site and block
updating, regardless of whether the data are on a regular or an irregular
lattice. The approach provides a good compromise between statistical and
computational efficiency and is accessible to statisticians without expertise
in numerical analysis or advanced computing.
