Chentsov’s theorem for exponential families

Abstract

Chentsov’s theorem characterizes the Fisher information metric on statistical models as the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. This implies that each statistical model is equipped with a natural 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 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. We then extend this result to curved exponential families. Our approach is based on the central limit theorem, so it gives a unified proof for discrete and continuous exponential families, and it is less technical than previous approaches.