Summary.We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure p D for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general p D approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the 'hat' matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding p D to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

On the 24 th November 2021 the sequence of a new SARS CoV-2 viral isolate Omicron-B.1.1.529 was announced, containing far more mutations in Spike (S) than previously reported variants. Neutralization titres of Omicron by sera from vaccinees and convalescent subjects infected with early pandemic as well as Alpha, Beta, Gamma, Delta are substantially reduced or fail to neutralize. Titres against Omicron are boosted by third vaccine doses and are high in cases both vaccinated and infected by Delta. Mutations in Omicron knock out or substantially reduce neutralization by most of a large panel of potent monoclonal antibodies and antibodies under commercial development. Omicron S has structural changes from earlier viruses, combining mutations conferring tight binding to ACE2 to unleash evolution driven by immune escape, leading to a large number of mutations in the ACE2 binding site which rebalance receptor affinity to that of early pandemic viruses.

Summary The essentials of our paper of 2002 are briefly summarized and compared with other criteria for model comparison. After some comments on the paper's reception and influence, we consider criticisms and proposals forimprovement made by us and others.

Gibbs sampling is a powerful technique for statistical inference. It involves little more than sampling from full conditional distributions, which can be both complex and computationally expensive to evaluate. Gilks and Wild have shown that in practice full conditionals are often logconcave, and they proposed a method of adaptive rejection sampling for efficiently sampling from univariate log-concave distributions. In this paper, to deal with non-log-concave full conditional distributions, we generalize adaptive rejection sampling to include a Hastings-Metropolis algorithm step. One important field of application in which statistical models may lead to non-log-concave full conditionals is population pharmacokinetics. Here, the relationship between drug dose and blood or plasma concentration in a group of patients typically is modelled by using non-linear mixed effects models. Often, the data used for analysis are routinely collected hospital measurements, which tend to be noisy and irregular. Consequently, a robust (t-distributed) error structure is appropriate to account for outlying observations andlor patients. We propose a robust non-linear full probability model for population pharmacokinetic data. We demonstrate that our method enables Bayesian inference for this model, through an analysis of antibiotic administration in new-born babies.

The study of spatial variations in disease rates is a common epidemiological approach used to describe the geographical clustering of diseases and to generate hypotheses about the possible`causes' which could explain apparent differences in risk. Recent statistical and computational developments have led to the use of realistically complex models to account for overdispersion and spatial correlation. However, these developments have focused almost exclusively on spatial modelling of a single disease. Many diseases share common risk factors (smoking being an obvious example) and, if similar patterns of geographical variation of related diseases can be identi®ed, this may provide more convincing evidence of real clustering in the underlying risk surface. We propose a shared component model for the joint spatial analysis of two diseases. The key idea is to separate the underlying risk surface for each disease into a shared and a disease-speci®c component. The various components of this formulation are modelled simultaneously by using spatial cluster models implemented via reversible jump Markov chain Monte Carlo methods. We illustrate the methodology through an analysis of oral and oesophageal cancer mortality in the 544 districts of Germany, 1986±1990.