Exponential (or other) families of distributions on manifolds.

Question

The exponential family is a general parametrized class of probability distributions on $R^n$ that has many nice properties (ML estimation among them) and includes most of the "standard" distributions one encounters (Gaussian, multinomial, exponential, $\chi^2$ etc).

Are there similarly well-defined parametrized families of distributions for manifold-valued random variables ? Specifically, if you have a general Riemannian manifold ? Or asked another way, is there an equivalent notion of an "exponential family" for a Riemannian manifold ?

Answer 1

The book Directional Statistics by Mardia and Jupp discusses concrete examples of distributions on:

1. Surface of the unit hypersphere
2. On Stiefel Manifolds
3. Maybe some others

EDIT In particular, have a look at §13.4.2 that discusses distributions on more general manifolds (e.g., on compact Riemannian manifolds). That section also provides several useful references.

I also recalled another source that might be useful to you:

Matrix Variate Distributions by Gupta and Nagar. In particular, see chapter 8.

Answer 2

There are several subtle aspects to these questions that I am sure have emerged for you since you first posted this question. The primary one is of course that probability spaces are assumed to have algebraic structures so that moments and other elements for statistical investigation of a random variable can be found -- or at least defined -- through direct analytic methods.

For example, the kth moment of a random variable with values in $(0,1)$ and probability distribution $f(x)$ is $\int_0^1 x^k f(x) dx.$ Note that you need to be able to multiply the values of the random variables to themselves and a real-valued function ($f$). Even multivariate statistics (which lacks multiplication) utilizes the vector-algebraic structure of $R^N$ to calculate moments etc.

All of this is lacking on a general manifold. Embedding the manifold into Euclidean space will not help, since you would typically get expected values, etc. living outside of the manifold, although it would be rather entertaining to tell someone that the expected location of the crash site of a satellite with decaying orbit is the center of the Earth.

Anyway, that's a long-winded introduction to point out that you should check out the following papers:
Pennec, X. "Probabilities and statistics on Riemannian manifolds: a geometric approach" http://hal.inria.fr/inria-00071490/

Pennec, X. "Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements." IEEE Workshop on Nonlinear Signal and Image Processing. Vol. 4. 1999.

Bochner in his text ${\it Harmonic\ Analysis\ and\ the\ Theory\ of\ Probability}$ describes stable distributions for general probability spaces, which is picked up for hyperbolic spaces in Getoor, R. K. "Infinitely divisible probabilities on the hyperbolic plane." Pacific Journal of Mathematics 11.4 (1961): 1287-1308.

Answer 3

I claim that the answer is yes. The key point that makes a distribution exponential is that there is a set of sufficient statistics, of the dimension of the manifold, that are additive.

On an $n$-dimensional manifold, chose $n$ functions $T_1$ through $T_n$ to $\mathbb R$. Given an $h(x)$ that is small when these functions are large, there is a unique $A(\eta)$ such that:

$\int h(x) e^{\sum_i \eta_i T_i - A(\eta)}=1$

That's the analogue of an exponential distribution.

Now, which $T_i$ and $h(x)$ should you pick? I don't know. It depends on the manifold.