Symmetric measures via moments
Alexey Koloydenko
(2004) Eurandom, Eindhoven, The Netherlands.

## Abstract

A finite $G\le GL(m,\mathbb{R})$ fixes $\Omega \subset \mathbb{R}^m$ and induces its action on $\mathcal{P}$, the set of probability distributions on $\Omega$. $\mathcal{P}^G$ is the set of distributions invariant under this action. We consider models based on $\mathcal{P}^G$. Ignoring the invariance, a common approach to modeling $P\in\mathcal{P}$ is to progressively match its moments. Among all distributions with a requested match, one reasonable choice is $P'$ that maximizes the entropy $H(P')$. Matching in the limit all the moments guarantees convergence to $P$ if $P$ is uniquely determined by its moments. We thereby generalize ordinary determinacy to determinacy within $\mathcal{P}^G$ and prove sufficiency of $G$-invariant moments for the latter. Using generators of $G$-invariant polynomials, we also give several sufficient conditions for the generalized property to hold. For applications, we propose a sequential procedure with adaptive convergence toward $P$. The procedure combines with one's favorite statistical model selection principle, and we present two such examples. We also describe a distribution of small subimages extracted from a large database of natural images, and compute generators for the relevant invariance. We discuss computations of $G$-invariant probability distributions. For example, concerned with computational efficiency, we lift the invariantly constrained entropy maximization problem to an appropriate quotient space of lower dimension''.