Biometrika Advance Access published online on February 28, 2007
Biometrika, doi:10.1093/biomet/asm017
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Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models
Graduate School of Business Administration, Keio University, 2-1-1 Hiyoshi-Honcho, Kohoku-ku, Yokohama-shi, Kanagawa, 223-8523, Japan
andoh{at}hc.cc.keio.ac.jp
Received for publication 1 September 2005.
Revision received 1 July 2006.
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Abstract |
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The problem of evaluating the goodness of the predictive distributions of hierarchical Bayesian and empirical Bayes models is investigated. A Bayesian predictive information criterion is proposed as an estimator of the posterior mean of the expected loglikelihood of the predictive distribution when the specified family of probability distributions does not contain the true distribution. The proposed criterion is developed by correcting the asymptotic bias of the posterior mean of the loglikelihood as an estimator of its expected loglikelihood. In the evaluation of hierarchical Bayesian models with random effects, regardless of our parametric focus, the proposed criterion considers the bias correction of the posterior mean of the marginal loglikelihood because it requires a consistent parameter estimator. The use of the bootstrap in model evaluation is also discussed.
Key Words: empirical Bayes model • hierarchical Bayesian model • Markov chain Monte Carlo • model misspecification