Laplace approximations and Bayesian information criteria in possibly misspecified models

We provide general conditions to ensure the valid Laplace approximations to the marginal likelihoods under model misspecification, and derive the Bayesian information criteria including all terms of order O-p(1). Under conditions in theorem 1 of Lv and Liu [J. R. Statist. Soc. B, 76, (2014), 141-167] and a continuity condition for prior densities, asymptotic expansions with error terms of order o(p)(1) are derived for the log-marginal likelihoods of possibly misspecified generalized linear models. We present some numerical examples to illustrate the finite sample performance of the proposed information criteria in misspecified models.