Generalizing the derivation of the Schwarz information criterion

The Schwarz information criterion (SIC, BIG, SEC) is one of the most widely known and used tools in statistical model selection. The criterion was derived by Schwarz (1978) to serve as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. Although the original derivation assumes that the observed data is independent, identically distributed, and arising from a probability distribution in the regular exponential family, SIC has traditionally been used in a much larger scope of model selection problems. To better justify the widespread applicability of SIG, we derive the criterion in a very general framework: one which does not assume any specific form for the likelihood function, but only requires that it satisfies certain non-restrictive regularity conditions.