Groupwise Bayesian Dimension Reduction

Nearly all existing estimations of the central subspace in regression take the frequentist approach. However, when the predictors fall naturally into a number of groups, these frequentist methods treat all predictors indiscriminately and can result in loss of the group-specific relation between the response and the predictors. In this article, we propose a Bayesian solution for dimension reduction which incorporates such group knowledge. We place a prior whose variance is constrained to the form of a direct sum on the central subspace and directly model the response density in terms of the sufficient predictors using a finite mixture model. This approach is computationally efficient and offers a unified framework to handle categorical predictors, missing predictors, and Bayesian variable selection. We illustrate the method using both a simulation study and an analysis of a temperature data set.