14 - Nonparametric Bayes regression, classification and hypothesis testing on manifolds

Summary

This chapter develops nonparametric Bayes procedures for classification, hypothesis testing and regression. The classification of a random observation to one of several groups is an important problem in statistics. This is the objective in medical diagnostics, the classification of subspecies, and, more generally, the target of most problems in image analysis. Equally important is the estimation of the regression function of Y given X and the prediction of Y given a random observation X. Here Y and X are, in general, manifold-valued, and we use nonparametric Bayes procedures to estimate the regression function.

Introduction

Consider the general problem of predicting a response Y ∈ Y based on predictors X ∈ X, where Y and X are initially considered to be arbitrary metric spaces. The spaces can be discrete, Euclidean, or even non-Euclidean manifolds. In the context of this book, such data arise in many chapters. For example, for each study subject, we may obtain information on an unordered categorical response variable such as the presence/absence of a particular feature as well as predictors having different supports including categorical, Euclidean, spherical, or on a shape space. In this chapter we extend the methods of Chapter 13 to define a very general nonparametric Bayes modeling framework for the conditional distribution of Y given X = x through joint modeling of Z = (X, Y). The flexibility of our modelling approach will be justified theoretically through Theorems, Propositions, and Corollaries 14.1, 14.2, 14.3, 14.4, and 14.5.