Assessment of two approximation methods for computing posterior model probabilities

Model selection is an important problem in statistical applications. Bayesian model averaging provides an alternative to classical model selection procedures and allows researchers to consider several models from which to draw inferences. In the multiple linear regression case, it is difficult to compute exact posterior model probabilities required for Bayesian model averaging. To reduce the computational burden the Laplace approximation and an approximation based on the Bayesian information criterion (BIC) have been proposed. The BIC approximation is the easiest to calculate and is being used widely in application. In this paper we conduct a simulation study to determine which approximation performs better. We give an example of where the methods differ, study the performance of these methods on randomly generated models and explore some of the features of the approximations. Our simulation study suggests that the Laplace approximation performs better on average than the BIC approximation.