A Variant of AIC Based on the Bayesian Marginal Likelihood

We propose information criteria that measure the prediction risk of a predictive density based on the Bayesian marginal likelihood from a frequentist point of view. We derive criteria for selecting variables in linear regression models, assuming a prior distribution of the regression coefficients. Then, we discuss the relationship between the proposed criteria and related criteria. There are three advantages of our method. First, this is a compromise between the frequentist and Bayesian standpoints because it evaluates the frequentist’s risk of the Bayesian model. Thus, it is less influenced by a prior misspecification. Second, the criteria exhibits consistency when selecting the true model. Third, when a uniform prior is assumed for the regression coefficients, the resulting criterion is equivalent to the residual information criterion (RIC) of Shi and Tsai (J. R. Stat. Soc. Ser. B64, 237–252 2002).