Bayesian Model Averaging Sufficient Dimension Reduction

Abstract

In sufficient dimension reduction (Li, 1991; Cook, 1998b), original predictors are replaced by their low-dimensional linear combinations while preserving all of the conditional information of the response given the predictors. Sliced inverse regression [SIR; Li, 1991] and principal Hessian directions [PHD; Li, 1992] are two popular sufficient dimension reduction methods, and both SIR and PHD estimators involve all of the original predictor variables. To deal with the cases when the linear combinations involve only a subset of the original predictors, we propose a Bayesian model averaging (Raftery et al., 1997) approach to achieve sparse sufficient dimension reduction. We extend both SIR and PHD under the Bayesian framework. The superior performance of the proposed methods is demonstrated through extensive numerical studies as well as a real data analysis.