A note on consistency of Bayesian high-dimensional variable selection under a default prior

Zellner's (Formula presented.) -prior is one of the most popular choices for model selection in Bayesian linear regression. Despite its popularity, the asymptotic theory for high-dimensional variable selection is not yet fully developed. In this paper, we investigate the asymptotic behaviour of Bayesian model selection under the (Formula presented.) -prior as the model dimension grows with the sample size. We find a simple and intuitive condition under which the posterior model distribution tends to be concentrated on the true model as the sample size increases even if the number of predictors grows much faster than the sample size does. Simulation study results indicate that satisfaction of this condition is essential for the success of Bayesian high-dimensional variable selection under the (Formula presented.) -prior.