A design-based approximation to the Bayes Information Criterion in finite population sampling

In this article, various issues related to the implementation of the usual Bayesian Information Criterion (BIC) are critically examined in the context of modelling a finite population. A suitable design-based approximation to the BIC is proposed in order to avoid the derivation of the exact likelihood of the sample which is often very complex in a finite population sampling. The approximation is justified using a theoretical argument and a Monte Carlo simulation study.