Statistica Sinica 17(2007), 667-690

ADAPTIVE BAYESIAN CRITERIA IN VARIABLE

SELECTION FOR GENERALIZED LINEAR MODELS

Xinlei Wang and Edward I. George

Southern Methodist University and University of Pennsylvania

Abstract: For the problem of variable selection in generalized linear models, we develop various adaptive Bayesian criteria. Using a hierarchical mixture setup for model uncertainty, combined with an integrated Laplace approximation, we derive Empirical Bayes and Fully Bayes criteria that can be computed easily and quickly. The performance of these criteria is assessed via simulation and compared to other criteria such as $AIC$ and $BIC$ on normal, logistic and Poisson regression model classes. A Fully Bayes criterion based on a restricted region hyperprior seems to be the most promising. Finally, our criteria are illustrated and compared with competitors on a data example.

Key words and phrases: AIC, BIC, empirical Bayes, fully Bayes, hierarchical Bayes, Laplace approximation.