Information geometry and sufficient statistics

TL;DR

This paper extends the geometric understanding of statistical models by analyzing how key structures like the Fisher metric and Amari-Chentsov tensor behave under sufficient statistics in infinite sample scenarios, addressing topological challenges.

Contribution

It introduces a framework for tensor fields on parametrized measure models that preserves geometric structures under sufficient statistics in infinite sample contexts.

Findings

Fisher metric and Amari-Chentsov tensor are uniquely characterized by invariance under sufficient statistics.

The framework handles topological issues in infinite sample size cases.

A decomposition of Markov morphisms and a monotonicity result for Fisher information are established.

Abstract

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari-Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.