Accelerating computation in Markov random field models for spatial data via structured MCMC

Spatial Poisson models for areal count data use nonstationary "intrinsic autoregressions," also often referred to as "conditionally autoregressive" (CAR) models. Bayesian inference for these models has generally involved using single parameter updating Markov chain Monte Carlo algorithms, which often exhibit slow mixing (i.e., poor convergence) properties. These spatial models are richly parameterized and lend themselves to the structured Markov chain Monte Carlo (SMCMC) algorithms. SMCMC provides a simple, general, and flexible framework for accelerating convergence in an MCMC sampler by providing a systematic way to block groups of similar parameters while taking full advantage of the posterior correlation structure induced by the model and data. Among the SMCMC strategies considered here are blocking using different size blocks (grouping by geographical region), reparameterization, updating jointly with and without model hyperparameters, "oversampling" some of the model parameters, and "pilot adaptation" versus continuous tuning techniques for the proposal density. We apply the techniques presented here to datasets on cancer mortality and late detection in the state of Minnesota. We find that, compared to univariate sampling procedures, our techniques will typically lead to more accurate posterior estimates, and they are sometimes also far more efficient in terms of the number of effective samples generated per second.