<p>We propose nonparametric Bayesian models for supervised dimension</p><p>reduction and regression problems. Supervised dimension reduction is</p><p>a setting where one needs to reduce the dimensionality of the</p><p>predictors or find the dimension reduction subspace and lose little</p><p>or no predictive information. Our first method retrieves the</p><p>dimension reduction subspace in the inverse regression framework by</p><p>utilizing a dependent Dirichlet process that allows for natural</p><p>clustering for the data in terms of both the response and predictor</p><p>variables. Our second method is based on ideas from the gradient</p><p>learning framework and retrieves the dimension reduction subspace</p><p>through coherent nonparametric Bayesian kernel models. We also</p><p>discuss and provide a new rationalization of kernel regression based</p><p>on nonparametric Bayesian models allowing for direct and formal</p><p>inference on the uncertain regression functions. Our proposed models</p><p>apply for high dimensional cases where the number of variables far</p><p>exceed the sample size, and hold for both the classical setting of</p><p>Euclidean subspaces and the Riemannian setting where the marginal</p><p>distribution is concentrated on a manifold. Our Bayesian perspective</p><p>adds appropriate probabilistic and statistical frameworks that allow</p><p>for rich inference such as uncertainty estimation which is important</p><p>for measuring the estimates. Formal probabilistic models with</p><p>likelihoods and priors are given and efficient posterior sampling</p><p>can be obtained by Markov chain Monte Carlo methodologies,</p><p>particularly Gibbs sampling schemes. For the supervised dimension</p><p>reduction as the posterior draws are linear subspaces which are</p><p>points on a Grassmann manifold, we do the posterior inference with</p><p>respect to geodesics on the Grassmannian. The utility of our</p><p>approaches is illustrated on simulated and real examples.</p> / Dissertation