  Variable selection in the linear regression model takes many apparent faces
from both frequentist and Bayesian standpoints. In this paper we introduce a
variable selection method referred to as a rescaled spike and slab model. We
study the importance of prior hierarchical specifications and draw connections
to frequentist generalized ridge regression estimation. Specifically, we study
the usefulness of continuous bimodal priors to model hypervariance parameters,
and the effect scaling has on the posterior mean through its relationship to
penalization. Several model selection strategies, some frequentist and some
Bayesian in nature, are developed and studied theoretically. We demonstrate the
importance of selective shrinkage for effective variable selection in terms of
risk misclassification, and show this is achieved using the posterior from a
rescaled spike and slab model. We also show how to verify a procedure's ability
to reduce model uncertainty in finite samples using a specialized forward
selection strategy. Using this tool, we illustrate the effectiveness of
rescaled spike and slab models in reducing model uncertainty.
