On Block Updating in Markov Random Field Models for Disease Mapping

Abstract

ABSTRACT. 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 poor 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 applications: 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. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields (Rue, 2001). By comparison, Monte Carlo estimates 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.

Efficient simulation and integrated likelihood estimation in state space models

“…First, this method is very efficient and has been widely used in nonparametric regression (see, for example, Silverman, 1985;Fahrmeir and Lang, 2001;Chib and Jeliazkov, 2006). Computational efficiency aspects of banded matrix operations are discussed in Golub and van Loan (1983), Knorr-Held and Rue (2002), Rue and Held (2005) and McCausland et al (2009). In the examples in Section 3, this algorithm generated 20-40% faster run times than the Kalman filter and smoother recursions.…”

“…The techniques are scalable in the dimension of the model and do not involve the Kalman filtering and smoothing recursions. The approach builds upon techniques that have been heavily used in nonparametric regression (e.g., Silverman, 1985;Fahrmeir and Lang, 2001;Chib and Jeliazkov, 2006;Chib et al, 2009), spatial models (Rue, 2001;Knorr-Held and Rue, 2002) and smooth coefficient models (Koop and Tobias, 2006) and although the applicability of these methods to state space models has been recognised (Fahrmeir and Kaufmann, 1991;Moura, 2002, 2005;Knorr-Held and Rue, 2002), they have not yet gained prominence in time-series analysis despite their versatility. Third, we show that the integrated likelihood ( | ), f y θ which gives the density of the data conditional on the parameters but integrated over the state vector , η can be obtained very easily.…”

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

“…To speed up convergence, it may be necessary to implement a joint move which updates θ and (α, β) jointly (Knorr- Held and Rue, 2002). We use the following construction.…”

“…Joint block updating of η and u, as proposed in Knorr- Held and Rue (2002), is based on the GMRF approximation as described in detail in Rue and Held (2005, Subsection 4.4.1). Basically a GMRF Metropolis-Hastings proposal is computed based on a quadratic Taylor approximation to the Poisson likelihood.…”

Specifying a Gaussian Markov Random Field by a Sparse Cholesky Triangle

“…GMRFs are also named as conditional autoregressive models (CARs) due to the seminal work of Besag (1974Besag ( , 1975 in spatial statistics. Intrinsic versions of GMRFs are also extensively used, see for example Besag and Higdon (1999), Fahrmeir and Lang (2001), Knorr-Held and Rue (2002) and Banerjee et al (2004).…”