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Model selection for regression problems with an increasing number ofcovariates continues to be an important problem both theoretically and inapplications. Model selection consistency and mean structure reconstructiondepend on the interplay between the Bayes factor learning rate and thepenalization on model complexity. In this work, we present results for theZellner-Siow prior for regression coefficients paired with a Poisson prior formodel complexity. We show that model selection consistency restricts thedimension of the true model from increasing too quickly. Further, we show thatthe additional contribution to the mean structure from new covariates must belarge enough to overcome the complexity penalty. The average Bayes factors fordifferent sets of models involves random variables over the choices of columnsfrom the design matrix. We show that a large class these random variables haveno moments asymptotically and need to be analyzed using stable laws. We derivethe domain of attraction for these random variables and obtain conditions onthe design matrix that provide for the control of false discoveries.

Revisiting High Dimensional Bayesian Model Selection for Gaussian Regression