The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.
"Estimating the Dimension of a Model." Ann. Statist. 6 (2) 461 - 464, March, 1978. https://doi.org/10.1214/aos/1176344136