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Bayesian measures of model complexity and fit


  • David J. Spiegelhalter
  • Nicola G. Best
  • Bradley P. Carlin
  • Angelika Van Der Linde


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 pD 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 pD 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 pD 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.

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  • David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
  • Handle: RePEc:bla:jorssb:v:64:y:2002:i:4:p:583-639
    DOI: 10.1111/1467-9868.00353

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    References listed on IDEAS

    1. Dimitris Fouskakis & David Draper, 2002. "Stochastic Optimization: a Review," International Statistical Review, International Statistical Institute, vol. 70(3), pages 315-349, December.
    2. George Casella & Christian P, Robert & Martin T, Wells, 2000. "Mixture Models, Latent Variables and Partitioned Importance Sampling," Working Papers 2000-03, Center for Research in Economics and Statistics.
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