  In the context of a high-dimensional linear regression model, we propose the
use of an empirical correlation-adaptive prior that makes use of information in
the observed predictor variable matrix to adaptively address high collinearity,
determining if parameters associated with correlated predictors should be
shrunk together or kept apart. Under suitable conditions, we prove that this
empirical Bayes posterior concentrates around the true sparse parameter at the
optimal rate asymptotically. A simplified version of a shotgun stochastic
search algorithm is employed to implement the variable selection procedure, and
we show, via simulation experiments across different settings and a real-data
application, the favorable performance of the proposed method compared to
existing methods.
