Abstract: In the context of a high-dimensional linear regression model, we propose an empirical correlation-adaptive prior that uses information in the observed predictor variable matrix to adaptively address high collinearity. We use this prior to determine whether the parameters associated with the correlated predictors should be shrunk together or kept apart. Under certain conditions, we prove that our empirical Bayes posterior concentrates at the optimal rate. Therefore the benefits of correlation-adaptation in finite samples can be achieved without sacrificing asymptotic optimality. A version of the shotgun stochastic search algorithm is employed to compute the posterior and facilitate variable selection. Finally we demonstrate our method's favorable performance compared with that of existing methods using real and simulated data examples, even in ultrahigh-dimensional settings.

Key words and phrases: Collinearity, empirical Bayes, posterior convergence rate, stochastic search, variable selection.