Supervised dimension reduction using Bayesian mixture modeling

We develop a Bayesian framework for supervised dimension reduction using a flexible nonparametric Bayesian mixture modeling approach. Our method retrieves the dimension reduction or d.r. subspace by utilizing a dependent Dirichlet process that allows for natural clustering for the data in terms of both the response and predictor variables. Formal probabilistic models with likelihoods and priors are given and efficient posterior sampling of the d.r. subspace can be obtained by a Gibbs sampler. As the posterior draws are linear subspaces which are points on a Grassmann manifold, we output the posteriormean d.r. subspace with respect to geodesics on the Grassmannian. The utility of our approach is illustrated on a set of simulated and real examples.