Gaussian Markov random field (GMRF) models are commonlyufz to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-siteuinglealgorithms have been considered. However, convergence and mixing properties ofsuD algorithms can be extremely poordu to strong dependencies ofparameters in the posteriordistribuQ84K In this paper, we propose variou block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standardfu6 conditionals, and can be applied in amoduzK fashion in a large nugef of di#erent scenarios. For illu##Kzf0 n we consider three di#erent applications: twoformu8Df0z3 for spatial modelling of a single disease (with andwithou additionaluditionalfL parameters respectively), and one formu## ion for the joint analysis of two diseases. TheresuKK indicate that the largest benefits are obtained ifparameters and the corresponding hyperparameter areuefz#L jointly in one large block. Implementation ofsuQ block algorithms is relatively easy usyf methods for fast sampling ofGaungf3 Markov random fields (Rus 2001). By comparison, Monte Carlo estimates based on single-siteungle-s can be rather misleading, even for very long rugfOu resuL6 may have wider relevance for efficient MCMCsimu6z8#f in hierarchical models with Markov random field components.

this paper, we consider multicategorical time-space data (Y it ; x it ; s i ); i = 1; : : : ; n; t = 1; : : : ; T; where the spatial location or site s i of individual i is given as an additional information. A typical example are monthly register data from the German Employment Office 1 for the years 1980-1995, where Y it is the employment status (e.g. unemployed, part time job, full time job, others) of individual i during month t and s i is the district in Germany where i has its domicile. Data from surveys on forest damage are a further example: Damage state Y it of tree i in year t, indicated by the defoliation degree, is measured in ordered categories (none to severe) and s i is the site of the tree on a lattice map. In both examples, covariates can be categorical or continuous, and possibly time-varying

We give an approach for using flow information from a system of wells to characterize hydrologic properties of an aquifer. In particular, we consider experiments where an impulse of tracer fluid is injected along with the water at the input wells and its concentration is recorded over time at the uptake wells. We focus on characterizing the spatially varying permeability field which is a key attribute of the aquifer for determining flow paths and rates for a given flow experiment. As is standard for estimation from such flow data, we make use of complicated subsurface flow code which simulates the fluid flow through the aquifer for a particular well configuration and aquifer specification, which includes the permeability field over a grid. This ill-posed problem requires that some regularity be imposed on the permeability field. Typically this is accomplished by specifying a stationary Gaussian process model for the permeability field. Here we use an intrinsically stationary Markov random field which compares favorably and offers some additional flexibility and computational advantages. Our interest in quantifying uncertainty leads us to take a Bayesian approach, using Markov chain Monte Carlo for exploring the high-dimensional posterior distribution. We demonstrate our approach with several examples.

This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimension n when the likelihood consists of mutually independent data. We demonstrate that a class of non-Gaussian approximations can be constructed for a wide range of likelihood models. They have the appealing properties that exact samples can be drawn from them, the normalisation constant is computable, and the computational complexity is only O(n 2 ) in the spatial case. The non-Gaussian approximations are refined versions of a Gaussian approximation. The latter serves well if the likelihood is near-Gaussian, but it is not sufficiently accurate when the likelihood is not near-Gaussian or if n is large. The accuracy of our approximations can be tuned by intuitive parameters to near any precision. We apply

this article is to present hierarchical Bayesian approaches that allow to simultaneously incorporate temporal and spatial dependencies between pixels directly in the model formulation. For reasons of computational feasibility, models have to be comparatively parsimonious, without oversimplifying. We introduce parametric and semiparametric spatial and spatio-temporal models that proved appropriate and illustrate their performance by application to fMRI data from a visual stimulation experiment.

Disease incidence or mortality data are typically available as rates or counts for specified regions, collected over time. We propose Bayesian nonparametric spatial modeling approaches to analyze such data. We develop a hierarchical specification using spatial random effects modeled with a Dirichlet process prior. The Dirichlet process is centered around a multivariate normal distribution. This latter distribution arises from a log-Gaussian process model that provides a latent incidence rate surface, followed by block averaging to the areal units determined by the regions in the study. With regard to the resulting posterior predictive inference, the modeling approach is shown to be equivalent to an approach based on block averaging of a spatial Dirichlet process to obtain a prior probability model for the finite dimensional distribution of the spatial random effects. We introduce a dynamic formulation for the spatial random effects to extend the model to spatio-temporal settings. Posterior inference is implemented through Gibbs sampling. We illustrate the methodology with simulated data as well as with a data set on lung cancer incidences for all 88 counties in the state of Ohio over an observation period of 21 years.

We apply a generalized Bayesian age-period-cohort (APC) model to a dataset on lung cancer mortality in West Germany, 1952-1996. Our goal is to predict future deaths rates until the year 2010, separately for males and females. Since age and period is not measured on the same grid, we propose a generalized APC model where consecutive cohort parameters represent strongly overlapping birth cohorts. This approach results in a rather large number of parameters, where standard algorithms for statistical inference by Markov chain Monte Carlo methods turn out to be computationally intensive. We propose a more ecient implementation based on ideas of block sampling from the time series literature. We entertain two dierent formulations, penalizing either rst or second dierences of age, period and cohort parameters. To assess the predictive quality of both formulations, we rst forecast the rates for the period 1987-1996 based on data until 1986. A comparison with the actual observed rates is ma...

We apply a generalized Bayesian age-period-cohort (APC) model to a dataset on lung cancer mortality in West Germany, 1952-1996. Our goal is to predict future deaths rates until the year 2010, separately for males and females. Since age and period is not measured on the same grid, we propose a generalized APC model where consecutive cohort parameters represent strongly overlapping birth cohorts. This approach results in a rather large number of parameters, where standard algorithms for statistical inference by Markov chain Monte Carlo methods turn out to be computationally intensive. We propose a more ecient implementation based on ideas of block sampling from the time series literature. We entertain two dierent formulations, penalizing either rst or second dierences of age, period and cohort parameters. To assess the predictive quality of both formulations, we rst forecast the rates for the period 1987-1996 based on data until 1986. A comparison with the actual observed rates is ma...

This paper considers small-area estimation with proportion data, and discusses the choice of upper-level model for the variation over areas. Inference about the random e#ects for the areas may depend strongly on the choice of this model, but this choice is not a straightforward matter. We show that posterior distributions of the deviances for the competing models provide a valuable tool for this purpose, and for the model averaging needed when several models fit equally well. We illustrate the approach with a well-known data set, and contrast it with the deviance information criterion approach

selection for a class of spatio-temporal models for areal data