Sampling Strategies for Fast Updating of Gaussian Markov Random Fields

Abstract

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.

Keywords:

• Bayesian computation
• Cholesky factorization
• Chromatic Gibbs sampling
• Conditional autoregressive model
• Graph coloring
• Markov chain Monte Carlo

Supplementary Materials

The supplementary material contains additional figures and algorithms referred to in the text, as well as details concerning the posterior sampling algorithm used in the election example. The R code for implementing the numerical examples is also available as a supplement.

Acknowledgments

The authors thank the editor, an associate editor, and two anonymous referees for comments and suggestions that improved this work.

Additional information

Funding

This material is based upon work partially supported by the National Science Foundation (NSF) under grant DMS-1127974 to the Statistical and Applied Mathematical Sciences Institute. DAB is partially supported by NSF grants CMMI-1563435, EEC-1744497, and OIA-1826715. CSM is partially supported by National Institutes of Health grant R01 AI121351 and NSF grant OIA-1826715.