  The method of Bayesian variable selection via penalized credible regions
separates model fitting and variable selection. The idea is to search for the
sparsest solution within the joint posterior credible regions. Although the
approach was successful, it depended on the use of conjugate normal priors.
More recently, improvements in the use of global-local shrinkage priors have
been made for high-dimensional Bayesian variable selection. In this paper, we
incorporate global-local priors into the credible region selection framework.
The Dirichlet-Laplace (DL) prior is adapted to linear regression. Posterior
consistency for the normal and DL priors are shown, along with variable
selection consistency. We further introduce a new method to tune
hyperparameters in prior distributions for linear regression. We propose to
choose the hyperparameters to minimize a discrepancy between the induced
distribution on R-square and a prespecified target distribution. Prior
elicitation on R-square is more natural, particularly when there are a large
number of predictor variables in which elicitation on that scale is not
feasible. For a normal prior, these hyperparameters are available in closed
form to minimize the Kullback-Leibler divergence between the distributions.
