Robust Bayesian variable selection with sub-harmonic priors

This paper studies Bayesian variable selection in linear models with spherically symmetric error distributions. We give a series of proper prior distributions which converge in a certain sense to an improper prior distribution and for which the Bayes factor for each possible sub-model converges to the Bayes factor for the improper prior. This convergence justifies the use of the improper prior in variable selection. We also show that the resulting improper Bayes factors are independent of the particular sampling model when all sub-models are assumed to have the same error distribution. This gives a surprising robustness to the procedure which is analogous to that observed in certain Bayes estimation problems involving spherically symmetric error distributions. We also show that our procedure has model selection consistency as the sample size increases for fixed maximum number of predictors uniformly over the entire class of spherical error distributions. A simulation study indicates that the procedure performs well and stably relative to a BIC based alternative.