Nonparametric Bayesian Models for Supervised Dimension Reduction and Regression

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2009

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Abstract

We propose nonparametric Bayesian models for supervised dimension

reduction and regression problems. Supervised dimension reduction is

a setting where one needs to reduce the dimensionality of the

predictors or find the dimension reduction subspace and lose little

or no predictive information. Our first method retrieves the

dimension reduction subspace in the inverse regression framework by

utilizing a dependent Dirichlet process that allows for natural

clustering for the data in terms of both the response and predictor

variables. Our second method is based on ideas from the gradient

learning framework and retrieves the dimension reduction subspace

through coherent nonparametric Bayesian kernel models. We also

discuss and provide a new rationalization of kernel regression based

on nonparametric Bayesian models allowing for direct and formal

inference on the uncertain regression functions. Our proposed models

apply for high dimensional cases where the number of variables far

exceed the sample size, and hold for both the classical setting of

Euclidean subspaces and the Riemannian setting where the marginal

distribution is concentrated on a manifold. Our Bayesian perspective

adds appropriate probabilistic and statistical frameworks that allow

for rich inference such as uncertainty estimation which is important

for measuring the estimates. Formal probabilistic models with

likelihoods and priors are given and efficient posterior sampling

can be obtained by Markov chain Monte Carlo methodologies,

particularly Gibbs sampling schemes. For the supervised dimension

reduction as the posterior draws are linear subspaces which are

points on a Grassmann manifold, we do the posterior inference with

respect to geodesics on the Grassmannian. The utility of our

approaches is illustrated on simulated and real examples.

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Mao, Kai (2009). Nonparametric Bayesian Models for Supervised Dimension Reduction and Regression. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/1581.

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