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Zbl 1067.62010
Spiegelhalter, David J.; Best, Nicola G.; Carlin, Bradley P.; van der Linde, Angelika
Bayesian measures of model complexity and fit (with discussion). (English)
[J] J. R. Stat. Soc., Ser. B, Stat. Methodol. 64, No. 4, 583-639 (2002). ISSN 1369-7412; ISSN 1467-9868/e

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.

MSC 2000:: *62C10 Bayesian problems
62B10 Statistical information theory
62F15 Bayesian inference
62A01 Foundational and philosophical topics

Keywords: Bayesian model comparison; deviance information criterion; effective number of parameters; hierarchical models; leverage; Markov chain Monte Carlo methods; model dimension

Cited in: Zbl 1237.62010 Zbl 1181.62103 Zbl 1219.62114 Zbl 1112.62017 Zbl 1087.62127 Zbl 1083.62028

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Zentralblatt MATH Berlin [Germany]