A Variant of AIC Using Bayesian Marginal Likelihood

We propose an information criterion which measures the prediction risk of the predictive density based on the Bayesian marginal likelihood from a frequentist point of view. We derive the criteria for selecting variables in linear regression models by putting the prior on the regression coefficients, and discuss the relationship between the proposed criteria and other related ones. There are three advantages of our method. Firstly, this is a compromise between the frequentist and Bayesian standpoint because it evaluates the frequentist's risk of the Bayesian model. Thus it is less influenced by prior misspecification. Secondly, non-informative improper prior can be also used for constructing the criterion. When the uniform prior is assumed on the regression coefficients, the resulting criterion is identical to the residual information criterion (RIC) of Shi and Tsai (2002). Lastly, the criteria have the consistency property for selecting the true model.