We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. We follow Dempster in examining the posterior distribution of the log-likelihood under each model, from which we derive measures of fit and complexity (the effective number of parameters). These may be combined into a Deviance Information Criterion (DIC), which is shown to have an approximate decision-theoretic justification. Analytic and asymptotic identities reveal the measure of complexity to be a generalisation of a wide range of previous suggestions, with particular reference to the neural network literature. The contributions of individual observations to fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. The procedure is illustrated in a number of examples, and throughout it is emphasised that the required quantities are trivial to compute in a Markov chain Monte Carlo analysis, and require no analytic work for new...

Abstract. Arguments are presented to support increased emphasis on logical aspects of formal methods of analysis, depending on probability in the sense of R. A. Fisher. Formulating probabilistic models that convey uncertain knowledge of objective phenomena and using such models for inductive reasoning are central activities of individuals that introduce limited but necessary subjectivity into science. Statistical models are classified into overlapping types called here empirical, stochastic and predictive, all drawing on a common mathematical theory of probability, and all facilitating statements with logical and epistemic content. Contexts in which these ideas are intended to apply are discussed via three major examples. Key words and phrases: Logicism and proceduralism; specificity of analysis; formal subjective probability; complementarity; subjective and objective; formal and informal; empirical, stochastic and predictive models; U.S. national census; screening for chronic disease; global climate change.

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This paper considers small-area estimation with proportion data, and discusses the choice of upper-level model for the variation over areas. Inference about the random e#ects for the areas may depend strongly on the choice of this model, but this choice is not a straightforward matter. We show that posterior distributions of the deviances for the competing models provide a valuable tool for this purpose, and for the model averaging needed when several models fit equally well. We illustrate the approach with a well-known data set, and contrast it with the deviance information criterion approach

This paper provides a brief review of the more popular methods for comparing models in a Bayesian framework. Personal experience in implementing these methods in problems requiring mixture models is also referenced. 1