  In the problem of selecting variables in a multivariate linear regression
model, we derive new Bayesian information criteria based on a prior mixing a
smooth distribution and a delta distribution. Each of them can be interpreted
as a fusion of the Akaike information criterion (AIC) and the Bayesian
information criterion (BIC). Inheriting their asymptotic properties, our
information criteria are consistent in variable selection in both the
large-sample and the high-dimensional asymptotic frameworks. In numerical
simulations, variable selection methods based on our information criteria
choose the true set of variables with high probability in most cases.
