Bayes factor consistency for nested linear models with a growing number of parameters

• We propose an explicit closed-form Bayes factor without integral representation.
• We study the model selection consistency of the Bayes factor as model dimension grows.
• The consistency of the Bayes factor is compared with the one of the intrinsic Bayes factor.

In this paper, we consider the Bayesian approach to the model selection problem for nested linear regression models. Common Bayesian procedures to this problem are based on Zellner's g-prior with a hyper-prior for the scaling factor g. Maruyama and George (2011) recently adopted this procedure with the beta-prime distribution for g and derived an explicit closed-form Bayes factor without integral representation which is thus easy to compute. In addition, they have studied its corresponding model selection consistency for fixed number of parameters. Over recent years, linear regression models with a growing number of unknown parameters have gained increased popularity in practical applications, such as the clustering problem. This observation motivates us to further investigate the consistency of Bayes factor with the beta-prime distribution for g under a scenario in which the number of parameters increases with the sample size. Finally, the results presented here are compared with the ones for the Bayes factor under intrinsic priors in relevant literature.