Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo (MCMC) simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as usual covariates with fixed effects, metrical covariates with nonlinear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates are all treated within the same general framework by assigning appropriate priors with different forms and degrees of smoothness. The approach is particularly appropriate for discrete and other fundamentally nonGaussian responses, where Gibbs sampling techniques developed for Gaussian m...

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.

P-splines are an attractive approach for modelling nonlinear smooth effects of covariates within the generalized additive and varying coefficient models framework. In this paper we propose a Bayesian version for P-splines and generalize the approach for one dimensional curves to two dimensional surface fitting for modelling interactions between metrical covariates. A Bayesian approach to P-splines has the advantage of allowing for simultaneous estimation of smooth functions and smoothing parameters. Moreover, it can easily be extended to more complex formulations, for example to mixed models with random effects for serially or spatially correlated response. Additionally, the assumption of constant smoothing parameters can be replaced by allowing the smoothing parameters to be locally adaptive. This is particularly useful in situations with changing curvature of the underlying smooth function or where the function is highly oscillating. Inference is fully Bayesian and uses recent MCMC techniques for drawing random samples from the posterior. In a couple of simulation studies the performance of Bayesian P-splines is studied and compared to other approaches in the literature. We illustrate the approach by a complex application on rents for flats in Munich.

Dynamic models extend state space models to non-normal observations. This paper suggests a specific hybrid Metropolis-Hastings algorithm as a simple device for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unknown, time-varying parameters are used to update the corresponding full conditional distributions. It is shown through simulated examples that the methodology has optimal performance in situations where the prior is relatively strong compared to the likelihood. Typical examples include smoothing priors for categorical data. A specific blocking strategy is proposed to ensure good mixing and convergence properties of the simulated Markov chain. It is also shown that the methodology is easily extended to robust transition models using mixtures of normals. The applicability is illustrated with an analysis of a binomial and a binary time series, known in the literature.

This paper combines existing models for longitudinal and spatial data in a hierarchical Bayesian framework, with particular emphasis on the role of time-- and space-- varying covariate effects. Data analysis is implemented via Markov chain Monte Carlo methods. The methodology is illustrated by a tentative re-analysis of Ohio lung cancer data 1968-88. Two approaches that adjust for unmeasured spatial covariates, particularly tobacco consumption, are described. The first includes random effects in the model to account for unobserved heterogeneity; the second adds a simple urbanization measure as a surrogate for smoking behaviour. The Ohio dataset has been of particular interest because of the suggestion that a nuclear facility in the southwest of the state may have caused increased levels of lung cancer there. However, we contend here that the data are inadequate for a proper investigation of this issue. Email: leo@stat.uni-muenchen.de 1 Introduction Data on disease incidence or mor...

Introduction Site specic farming is aiming at targeting inputs of fertiliser, pesticide, and herbicide according to locally determined requirements. In connection with herbicide application on a eld, it is important to map the weed intensity so that the dose of herbicide applied at any location can be adjusted to the amount of weed present at the location. In a Danish project on precision farming (Olesen, 1997) one objective was to investigate whether observations of soil properties could be used for prediction of weed intensity. In practice the farmer or his advisor should then establish a relation between soil properties and weed occurrence from extensive observations collected one year and use this for prediction of the weed intensity in subsequent years where only a limited number of weed count observations would be 1 collected. Many soil properties are fairly constant over time so that observations of soil samples obtained the rst year can also be used in subseq

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

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This chapter describes a nonparametric Bayesian approach to the estimation of curves and surfaces that act as parameters in statistical models. The approach is based on mixing variable dimensional piecewise constant approximations, whose `smoothness' is regulated by a Markov random field prior. Random partitions of the domain are defined by Voronoi tessellations of random generating point patterns. Variable dimension Markov chain Monte Carlo methods are proposed for the numerical estimation, and a detailed algorithm is specified for one special case. General applicability of the approach is discussed in the context of density estimation, regression and interpolation problems, and an application to the intensity estimation for a spatial Poisson point process is presented.

This paper describes and illustrates a Bayesian approach to the statistical analysis of agricultural field experiments. Our basic formulation assumes an additive model, in which plot yields, possibly after transformation, are the aggregate of treatment (or variety) effects, fertility effects and measurement (technical) errors. However, this formulation can easily be modified to allow for discrete observations, as might occur in the assessment of quality or of pest resistance, without any need for continuous approximations. Both the treatment and the fertility effects are fixed quantities (parameters) and the former may be unstructured, as in an ordinary variety trial, or structured, as in a factoffal experiment. Randomness enters through the technical errors and through prior distributions for the treatment and fertility effects, chosen to reflect our opinions about them before the trial takes place. In particular, we adopt fertility priors that have a spatially dependent structure, since, as usual, we anticipate that the fertilities of plots close together are generally more similar than those of plots further apart. Not surprisingly, there are strong formal links with models used in classical spatial analysis of field experiments, though of course the interpretation here is quite different