Introduction This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some p-dimensional region of interest X . Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure between points to vary over the space of interest. Typically, the correlation between points is assumed to be a xed function which is most likely to be parameterised by a few variables that can be estimated from the data (see, for example, Diggle, Tawn and Moyeed (1998)). Partition models avoid the need for preexamination of the data to nd a suitable correlation function to use. This removes the bias necessarily introduced by picking the correlation function and estimating its parameters using the same set of data. Spatial clusters are, by their nature, regions which are not representative of the entire space of intere

this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.

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Constraints on the parameters in a Bayesian hierarchical model typically make Bayesian computation and analysis complicated. As Gelfand, Smith and Lee (1992) remarked, it is almost impossible to sample from a posterior distribution when its density contains analytically intractable integrals (normalizing constants) that depend on the (hyper) parameters. Therefore, the Gibbs sampler or the Metropolis algorithm can not be directly applied to such problems. In this paper, using the idea of "reweighting mixtures" of Geyer (1994), we develop alternative Monte Carlo based methods to determine properties of the desired Bayesian posterior distribution. Necessary theory and two illustrative examples are provided. Keywords and Phrases: Bayesian computation; Bayesian hierarchical model; Gibbs sampler; Markov chain Monte Carlo; Marginal posterior density estimation; Posterior distribution; Sensitivity of prior specification. 1 Introduction In this article we consider a Bayesian hierarchical mod...

this paper we try our hands at it. One version of the classical theory of factorial experiments, going back to Fisher and further developed by Kempthorne (1955), completely avoids distributional assumptions, assuming only additivity, and uses randomisation to derive the standard tests of hypotheses about treatment effects. Here, we are interested in the more familiar classical approach via linear modelling and normal distribution theory. The corresponding Bayesian analysis has been developed mainly in the pioneering works of Box & Tiao (1973) and Lindley & Smith (1972). Box & Tiao (1973, Chapter 6) discuss Bayesian analysis of cross classified designs, including fixed, random and mixed effects models. They point out that in a Bayesian approach the appropriate inference procedure for fixed and random effects "depends upon the nature of the prior distribution used to represent the behavior of the factors". They also show (Chapter 7) that shrinkage estimates of specific effects may result when a random effects model is assumed. Lindley & Smith (1972) use a hierarchically structured linear model built on multivariate normal components (special cases of the model are considered by Lindley, 1972 and Smith, 1973), with the focus on estimation of treatment effects. These are authoritative and attractive approaches, albeit with modest compromises to the Bayesian paradigm -- in respect of the estimation of the variance components -- necessitated by the computational limitations of the time. Nevertheless, the inference is almost entirely estimative: questions about the indistinguishability of factor levels, or more general hypotheses about contrasts, are answered indirectly trough their joint posterior distribution, e.g. by checking whether the hypothesis falls in a highest poster...

The determination of toxicokinetic parameters is an essential component in the risk assessment of potential harmful chemicals. It's a first step to analyse the processes which are involved in the development of DNA adducts and might therefore lead to the development of cancer. The complete research depends on investigations with animals in vivo and in vitro, so that a critical step is the extrapolation from experimental animals to the human organism. Besides the investigation of the interspecific differences, the intraspecific and the interoccasion variability have to be analysed to avoid serious errors in the determination of the human risk. The aim of extrapolation from one species to an other requires a characterisation of the interesting processes which is valid for the whole species, i.e. population mean parameters instead of sets of parameters for different individuals, occasions and concentrations of the interesting chemical. The theory of hierarchical models, basically the wor...

For modeling spatial processes, we propose rich classes of range anisotropic covariance structures that greatly increase the scope of variogram contours in R² and include geometric anisotropy and isotropy as special cases. We demonstrate how the class of all completely monotonic isotropic variograms can be extended to capture range anisotropy and illustrate with two examples, the Matérn and the general exponential. We adopt a Bayesian perspective and fit these range anisotropic covariance models using sampling-based methods. In the presence of measurement error/microscale effect, we develop the noiseless predictive distribution. We analyze a data set of scallop catches, withholding ten sites, to compare the accuracy and precision of the standard and noiseless predictive distributions.

In this paper we propose a new Monte Carlo EM algorithm to compute maximum likelihood estimates in the context of random effects models. The algorithm involves the construction of e cient sampling distributions for the Monte Carlo implementation of the E-step, together with a reweighting procedure that allows repeatedly using a same sample of random effects. In addition, we explore the use of stochastic approximations to speed up convergence once stability has been reached. Our algorithm is compared with that of McCulloch (1997). Extensions to more general problems are discussed.

Bayesian methods are developed for the multivariate nonparametric regression problem where the domain is taken to be a compact Riemannian manifold. In terms of the latter, the underlying geometry of the manifold induces certain symmetries on the multivariate nonparametric regression function. The Bayesian approach then allows one to incorporate hierarchical Bayesian methods directly into the spectral structure, thus providing a symmetry-adaptive multivariate Bayesian function estimator. One can also diffuse away some prior information in which the limiting case is a smoothing spline on the manifold. This, together with the result that the smoothing spline solution obtains the minimax rate of convergence in the multivariate nonparametric regression problem, provides good frequentist properties for the Bayes estimators. An application to astronomy is included.

While Bayes’ theorem has a 250-year history, and the method of inverse probability that flowed from it dominated statistical thinking into the twentieth century, the adjective “Bayesian” was not part of the statistical lexicon until relatively recently. This paper provides an overview of key Bayesian developments, beginning with Bayes’ posthumously published 1763 paper and continuing up through approximately 1970, including the period of time when “Bayesian” emerged as the label of choice for those who advocated Bayesian methods.