Polynomial invariants in statistical modelling

Alexey Koloydenko.

We discuss connections between statistics and analysis on one hand, and polynomial invariants of finite groups, on the other. Our motivation comes from studies of probability distributions of very small subimages (microimages) of digital images of complex scenes (e.g. photographs of landscapes, urban scenes, faces). These distributions appear to respect several symmetries that, when represented analytically, lead to polynomials with appropriate invariance. Let a (microimage) state space $\Omega \subset \mathbb{R}^m$ be fixed by $G$, a finite group of nonsingular linear transformations, and let $\mathcal{P}$ be probability distributions on $\Omega$. We consider $\mathcal{P}^G$, the set of probability distributions that are invariant under the induced action of $G$ on $\mathcal{P}$. Given evidence that $P\in \mathcal{P}$ is (approximately) $G$-invariant, we model $P$ by members of $\mathcal{P}^G$. In the absence of the $G$-invariance (i.e. $G$ consists of the identity transformation only), a distribution is commonly modeled by matching several of its (empirical) moments. Among all distributions that provide the requested match, one chooses $P'$ that maximizes the {\em entropy} $H(P)$. (Equivalently, $P'$ maximizes the likelihood under the exponential models in which the chosen moments provide a sufficient statistics.) Such moment pursuit is theoretically well-grounded when the modeled measure is determinate by its moments, the latter being exactly the subject of the ``problem of moments''. Since arbitrary moments of $P$ need not be $G$-invariant, we introduce {\em invariant moments} and {\em determinacy by invariant moments} and give appropriate sufficient conditions for the latter. This generalizes the ordinary multidimensional moment problem to invariant measures. We also introduce minimal generating subsets of invariant moments. To make these ideas useful in practice, we propose a sequential procedure with adaptive convergence toward $P$. This procedure can be used with one's favorite model selection principle, and we outline two concrete such settings. For our motivating example of the microimage distributions, we compute a set of generators for the relevant invariance, analyze the orbit space, and discuss relevant computational issues.

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