  Model selection is of fundamental importance to high dimensional modeling
featured in many contemporary applications. Classical principles of model
selection include the Kullback-Leibler divergence principle and the Bayesian
principle, which lead to the Akaike information criterion and Bayesian
information criterion when models are correctly specified. Yet model
misspecification is unavoidable when we have no knowledge of the true model or
when we have the correct family of distributions but miss some true predictor.
In this paper, we propose a family of semi-Bayesian principles for model
selection in misspecified models, which combine the strengths of the two
well-known principles. We derive asymptotic expansions of the semi-Bayesian
principles in misspecified generalized linear models, which give the new
semi-Bayesian information criteria (SIC). A specific form of SIC admits a
natural decomposition into the negative maximum quasi-log-likelihood, a penalty
on model dimensionality, and a penalty on model misspecification directly.
Numerical studies demonstrate the advantage of the newly proposed SIC
methodology for model selection in both correctly specified and misspecified
models.
