Abstract This chapter describes a Bayesian hierarchical model and applies it to a new dataset on insulin-dependent diabetes mellitus (IDDM) prevalence among 18-year-old males born in Sardinia between 1936 and 1973, using malaria prevalence in 1938–40 as the ecological covariate. It shows how to deal with the potential bias associated with using such a proxy by extending the Bayesian model to allow for covariate measurement error. It provides a method for choosing the hyperprior distributions for the spatial variation parameters of the model, and discusses the sensitivity of the results to different choices.

Markov chain Monte Carlo (MCMC) is used for evaluating expectations of functions of interest under a target distribution π. This is done by calculating averages over the sample path of a Markov chain having π as its stationary distribution. For computational efficiency, the Markov chain should be rapidly mixing. This sometimes can be achieved only by careful design of the transition kernel of the chain, on the basis of a detailed preliminary exploratory analysis of π. An alternative approach might be to allow the transition kernel to adapt whenever new features of π are encountered during the MCMC run. However, if such adaptation occurs infinitely often, then the stationary distribution of the chain may be disturbed. We describe a framework, based on the concept of Markov chain regeneration, which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages.

In a family study of breast cancer, epidemiologists in Southern California increase the power for detecting a gene-environment interaction. In Gambia, a study helps a vaccination program reduce the incidence of Hepatitis B carriage. Archaeologists in Austria place a Bronze Age site in its true temporal location on the calendar scale. And in France,

BACKGROUND:In recent years diagnostic practice in psychiatry has become increasingly structured in an attempt to standardize definitions of disorders and improve reliability. At the same time there has been an increasing recognition of the need to take account of uncertainty in the process of diagnostic decision making. For the most part, diagnosis is still represented by a binary outcome while this is known to entail a substantial loss of information. Many diagnostic schemes involve, in part, taking thresholds on the numbers of symptoms required from symptom lists.METHODS:A model is proposed here, using ideas derived from latent class analysis to permit generalization from these schemes through moving from a binary to a probabilistic measure of psychiatric case status and replacing thresholds with smoothed transitions.RESULTS:An outcome measure is produced where disorder status is expressed in terms of probabilities without changing the meaning of the original measure. Prevalence estimates (using ICD-10 Depressive Episode criteria) are more stable and can be given with increased precision.CONCLUSIONS:Disorder status when expressed in this way retains more diagnostic information and provides a useful extension to traditional binary analyses when looking at prevalence and risk factor estimation.

This paper considers the problem of scaling the proposal distribution of a multidimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to infinity. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process.The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions.The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.

We describe Bayesian hierarchical-spatial models for disease mapping with imprecisely observed ecological covariates. We posit smoothing priors for both the disease submodel and the covariate submodel. We apply the models to an analysis of insulin Dependent Diabetes Mellitus incidence in Sardinia, with malaria prevalence as a covariate.

Abstract The algorithm of Metropolis et al. (1953) and its generalizations have been increasingly popular in computational physics and, more recently, statistics, for sampling from intractable multivariate distributions. Much recent research has been devoted to increasing the efficiency of simulation algorithms by altering the jumping rules for Metropolis-like algorithms. We study a very specific question: What are the most efficient symmetric jumping kernels for simulating a normal target distribution using the Metropolis algorithmã We provide a general theoretical result as the dimension of a class of canonical problems goes to ∞ and numerical approximations and simulations for low-dimensional Gaussian target distributions that show that the limiting results provide extremely accurate approximations in six and higher dimensions.

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Bayesian random effects models may be fitted using Gibbs sampling, but the Gibbs sampler can be slow mixing due to what might be regarded as lack of model identifiability. This slow mixing substantially increases the number of iterations required during Gibbs sampling. We present an analysis of data on immunity after Rubella vaccinations which results in a slow-mixing Gibbs sampler. We show that this problem of slow mixing can be resolved by transforming the random effects and then, if desired, expressing their joint prior distribution as a sequence of univariate conditional distributions. The resulting analysis shows that the decline in antibodies after Rubella vaccination is relatively shallow compared to the decline in antibodies which has been shown after Hepatitis B vaccination.

We consider the convergence of adaptive direction sampling, concentrating mainly on a special case, the "snooker algorithm" for which a powerful irreducibility result can be proved under extremely mild regularity conditions.