We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.

The Markov chain Monte Carlo (MCMC) method, as a computer-intensive statistical tool, has enjoyed an enormous upsurge in interest over the last few years. This paper provides a simple, comprehensive and tutorial review of some of the most common areas of research in this ®eld. We begin by discussing how MCMC algorithms can be constructed from standard buildingblocks to produce Markov chains with the desired stationary distribution. We also motivate and discuss more complex ideas that have been proposed in the literature, such as continuous time and dimension jumping methods. We discuss some implementational issues associated with MCMC methods. We take a look at the arguments for and against multiple replications, consider how long chains should be run for and how to determine suitable starting points. We also take a look at graphical models and how graphical approaches can be used to simplify MCMC implementation. Finally, we present a couple of examples, which we use as case-studies to highlight some of the points made earlier in the text. In particular, we use a simple changepoint model to illustrate how to tackle a typical Bayesian modelling problem via the MCMC method, before using mixture model problems to provide illustrations of good sampler output and of the implementation of a reversible jump MCMC algorithm.

Foot-and-mouth disease (FMD) in the UK provides an ideal opportunity to explore optimal control measures for an infectious disease. The presence of fine-scale spatio-temporal data for the 2001 epidemic has allowed the development of epidemiological models that are more accurate than those generally created for other epidemics and provide the opportunity to explore a variety of alternative control measures. Vaccination was not used during the 2001 epidemic; however, the recent DEFRA (Department for Environment Food and Rural Affairs) contingency plan details how reactive vaccination would be considered in future. Here, using the data from the 2001 epidemic, we consider the optimal deployment of limited vaccination capacity in a complex heterogeneous environment. We use a model of FMD spread to investigate the optimal deployment of reactive ring vaccination of cattle constrained by logistical resources. The predicted optimal ring size is highly dependent upon logistical constraints but is more robust to epidemiological parameters. Other ways of targeting reactive vaccination can significantly reduce the epidemic size; in particular, ignoring the order in which infections are reported and vaccinating those farms closest to any previously reported case can substantially reduce the epidemic. This strategy has the advantage that it rapidly targets new foci of infection and that determining an optimal ring size is unnecessary.

Summary. The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms. The second group of methods generalizes the reversible jump algorithm by using the so-called saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed. The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illustrate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling.

Much work has been published on the theoretical aspects of simulated annealing. This paper provides a brief overview of this theory and provides an introduction to the practical aspects of function optimization using this approach. Different implementations of the general simulated annealing algorithm are discussed, and two examples are used to illustrate the behaviour of the algorithm in low dimensions. A third example illustrates a hybrid approach, combining simulated annealing with traditional techniques.

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Background: While viruses have long been shown to capitalize on their limited genomic size by utilizing both strands of DNA or complementary DNA/RNA intermediates to code for viral proteins, it has been assumed that human retroviruses have all their major proteins translated only from the plus or sense strand of RNA, despite their requirement for a dsDNA proviral intermediate. Several studies, however, have suggested the presence of antisense transcription for both HIV-1 and HTLV-1. More recently an antisense transcript responsible for the HTLV-1 bZIP factor (HBZ) protein has been described. In this study we investigated the possibility of an antisense gene contained within the human immunodeficiency virus type 1 (HIV-1) long terminal repeat (LTR).