Sampling Strategies for Fast Updating of Gaussian Markov Random Fields

Gaussian Markov random fields (GMRFs) are popular for modeling temporal or spatial dependence in large areal datasets due to their ease of interpretation and computational convenience afforded by conditional independence and their sparse precision matrices needed for random variable generation. Using such models inside a Markov chain Monte Carlo algorithm requires repeatedly simulating random fields. This is a nontrivial issue, especially when the full conditional precision matrix depends on parameters that change at each iteration. Typically in Bayesian computation, GMRFs are updated jointly in a block Gibbs sampler or one location at a time in a single-site sampler. The former approach leads to quicker convergence by updating correlated variables all at once, while the latter avoids solving large matrices. Efficient algorithms for sampling Markov random fields have become the focus of much recent research in the machine learning literature, much of which can be useful to statisticians. We briefly review recently proposed approaches with an eye toward implementation for statisticians without expertise in numerical analysis or advanced computing. In particular, we consider a version of block sampling in which the underlying graph can be cut so that conditionally independent sites are all updated together. This algorithm allows a practitioner to parallelize the updating of a subset locations or to take advantage of `vectorized' calculations in a high-level language such as R. Through both simulation and real data application, we demonstrate computational savings that can be achieved versus both traditional single-site updating and block updating, regardless of whether the data are on a regular or irregular lattice. We argue that this easily-implemented sampling routine provides a good compromise between statistical and computational efficiency when working with large datasets.