Bayesian deviance the e ective number of parameters and the comparison of arbitrarily complex models

We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly de ned We follow Dempster in examining the posterior distribution of the log likelihood under each model from which we derive measures of t and complexity the e ective number of parameters These may be combined into a Deviance Information Criterion DIC which is shown to have an approximate decision theoretic justi cation Ana lytic 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 t 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 models