Chentsov's theorem for exponential families

Chentsov’s theorem characterizes the Fisher information metrics on statistical models as essentially the only Riemannian metrics that are invariant under sufficient statistics. This implies that statistical models are naturally equipped with geometric structures, 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.