Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models

The problem of evaluating the goodness of the predictive distributionsof hierarchical Bayesian and empirical Bayes models is investigated.A Bayesian predictive information criterion is proposed as anestimator of the posterior mean of the expected loglikelihoodof the predictive distribution when the specified family ofprobability distributions does not contain the true distribution.The proposed criterion is developed by correcting the asymptoticbias of the posterior mean of the loglikelihood as an estimatorof its expected loglikelihood. In the evaluation of hierarchicalBayesian models with random effects, regardless of our parametricfocus, the proposed criterion considers the bias correctionof the posterior mean of the marginal loglikelihood becauseit requires a consistent parameter estimator. The use of thebootstrap in model evaluation is also discussed.