Book contents
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
13 - Nonparametric Bayes inference on manifolds
Published online by Cambridge University Press: 05 May 2012
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
Summary
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.
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- Information
- Nonparametric Inference on ManifoldsWith Applications to Shape Spaces, pp. 156 - 181Publisher: Cambridge University PressPrint publication year: 2012