# Chentsov's theorem for exponential families

## TL;DR

This paper extends Chentsov's theorem to exponential families, showing the Fisher information metric is uniquely invariant under key statistical transformations, using a unified, less technical proof based on the central limit theorem.

## Contribution

It proves a version of Chentsov's theorem for exponential families, characterizing the Fisher information metric as the unique invariant Riemannian metric under specific transformations.

## Key findings

- Fisher information metric is uniquely invariant in exponential families
- Unified proof applicable to both discrete and continuous cases
- Simplifies previous approaches using the central limit theorem

## Abstract

Chentsov's theorem characterizes the Fisher information metric on statistical models as essentially the only Riemannian metric that is invariant under sufficient statistics. This implies that each statistical model is naturally equipped with a geometry, so Chentsov's theorem explains why many statistical properties can be described in geometric terms. However, despite being one of the foundational theorems of statistics, Chentsov's theorem has only been proved previously in very restricted settings or under relatively strong regularity and invariance assumptions. We therefore prove a version of this theorem for the important case of exponential families. In particular, we characterise the Fisher information metric as the only Riemannian metric (up to rescaling) on an exponential family and its derived families that is invariant under independent and identically distributed extensions and canonical sufficient statistics. Our approach is based on the central limit theorem, so it gives a unified proof for both discrete and continuous exponential families, and it is less technical than previous approaches.