  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 (2002).
