Bayesian inference in high-dimensional linear models using an empirical correlation-adaptive prior

In the context of a high-dimensional linear regression model, we propose use an empirical correlation-adaptive prior that makes information in 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, prove this Bayes posterior concentrates around true sparse parameter at optimal rate asymptotically. A simplified version shotgun stochastic search algorithm is employed implement selection procedure, and show, via simulation experiments across different settings real-data application, favorable performance proposed method compared existing methods.