In this chapter we adapt and extend the nonparametric density estimation procedures on Euclidean spaces to general (Riemannian) manifolds.

Introduction

So far in this book we have used notions of center and spread of distributions on manifolds to identify them or to distinguish between two or more distributions. However, in certain applications, other aspects of the distribution may also be important. The reader is referred to the data in Section 14.5.3 for such an example. Also, our inference method so far has been frequentist.

In this chapter and the next, we pursue different goals and a different route. Our approach here and in the next chapter will be nonparametric Bayesian, which involves modeling the full data distribution in a flexible way that is easy to work with. The basic idea will be to represent the unknown distribution as an infinite mixture of some known parametric distribution on the manifold of interest and then set a full support prior on the mixing distribution. Hence the parameters defining the distribution are no longer finite-dimensional but reside in the infinite-dimensional space of all probabilities. By making the parameter space infinite-dimensional, we ensure a flexible model for the unknown distribution and consistency of its estimate under mild assumptions. All these will be made rigorous through the various theorems we will encounter in the subsequent sections.

For a prior on the mixing distribution, a common choice can be the Dirichlet process prior (see Ferguson, 1973, 1974). We present a simple algorithm for posterior computations in Section 13.4.