Effects of Bayesian Model Selection on Frequentist Performances: An Alternative Approach

Abstract

It is quite common in statistical modeling to select a model and make inference as if the model had been known in advance; i.e. ignoring model selection uncertainty. The resulted estimator is called post-model selection estimator (PMSE) whose properties are hard to derive. Conditioning on data at hand (as it is usually the case), Bayesian model selection is free of this phenomenon. This paper is concerned with the properties of Bayesian estimator obtained after model selection when the frequentist (long run) performances of the resulted Bayesian estimator are of interest. The proposed method, using Bayesian decision theory, is based on the well known Bayesian model averaging (BMA)’s machinery; and outperforms PMSE and BMA. It is shown that if the unconditional model selection probability is equal to model prior, then the proposed approach reduces BMA. The method is illustrated using Bernoulli trials.