Title: Chentsov's theorem for exponential families,
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.