Approximations and consistency of Bayes factors as model dimension grows

Stone (J. Roy. Statist. Soc. Ser. B 41 (1979) 276) showed that BIC can fail to be asymptotically consistent. Note, however, that BIC was developed as an asymptotic approximation to Bayes factors between models, and that the approximation is valid only under certain conditions. The counterexample of Stone arises in situations in which BIC is not an adequate approximation. We develop some new approximations to Bayes factors, that are valid for the situation considered in Stone (1979) and discuss related issues of consistency.