  This paper studies Bayesian variable selection in linear models with general
spherically symmetric error distributions. We propose sub-harmonic priors which
arise as a class of mixtures of Zellner's g-priors for which the Bayes factors
are independent of the underlying error distribution, as long as it is in the
spherically symmetric class. Because of this invariance to spherically
symmetric error distribution, we refer to our method as a robust Bayesian
variable selection method. We demonstrate that our Bayes factors have model
selection consistency and are coherent. We also develop Laplace approximations
to Bayes factors for a number of recently studied mixtures of g-priors that
have recently appeared in the literature (including our own) for Gaussian
errors. These approximations, in each case, are given by the Gaussian Bayes
factor based on BIC times a simple rational function of the prior's
hyper-parameters and the R^2's for the respective models. We also extend model
selection consistency for several g-prior based Bayes factor methods for
Gaussian errors to the entire class of spherically symmetric error
distributions. Additionally we demonstrate that our class of sub-harmonic
priors are the only ones within a large class of mixtures of g-priors studied
in the literature which are robust in our sense. A simulation study and an
analysis of two real data sets indicates good performance of our robust Bayes
factors relative to BIC and to other mixture of g-prior based methods.
