On block updating in Markov random field models for disease mapping. (REVISED, May 2001)

Gaussian Markov random field (GMRF) models are commonly used to
model spatial correlation in disease mapping applications. For
Bayesian inference by MCMC, so far mainly single-site updating
algorithms have been considered. However, convergence and mixing
properties of such algorithms can be extremely bad due to strong
dependencies of parameters in the posterior distribution. In this
paper, we propose various block sampling algorithms in order to
improve the MCMC performance. The methodology is rather general,
allows for non-standard full conditionals, and can be applied in a
modular fashion in a large number of different scenarios. For
illustration we consider three different models: two formulations
for spatial modelling of a single disease (with and without
additional unstructured parameters respectively), and one
formulation for the joint analysis of two diseases. We apply the
proposed algorithms to two datasets known from the literature. The
results indicate that the largest benefits are obtained if
parameters and the corresponding hyperparameter are updated jointly
in one large block. In certain situations, even updating of all or
nearly all parameters in one block may be necessary. Implementation
of such block algorithms is surprisingly easy using methods for
fast sampling of Gaussian Markov random fields (Rue, 2000). By
comparison, estimates of the relative risk and related quantities,
such as the posterior probability of an exceedence relative risk,
based on single-site updating, can be rather misleading, even for
very long runs. Our results may have wider relevance for efficient
MCMC simulation in hierarchical models with Markov random field
components.