Uniqueness of the Fisher–rao Metric on the Space of Smooth Densities

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher–Rao metric. Introduction. The Fisher–Rao metric on the space Prob(M) of probability densities is of importance in the field of information geometry. Restricted to finitedimensional submanifolds of Prob(M), so-called statistical manifolds, it is called Fisher’s information metric [1]. The Fisher–Rao metric has the property that it is invariant under the action of the diffeomorphism group. The interesting question is whether it is the unique metric possessing this invariance property. A uniqueness result was established [4, p. 156] for Fisher’s information metric on finite sample spaces and [2] extended it to infinite sample spaces. The Fisher–Rao metric on the infinite-dimensional manifold of all positive probability densities was studied in [5], including the computation of its curvature. A consequence of our main theorem in this article is the infinite-dimensional analogue of the result in [4]: Theorem. Let M be a compact manifold without boundary of dimension ≥ 2. Then any smooth weak Riemannian metric on the space Prob(M) of smooth positive probability densities, that is invariant under the action of the diffeomorphism group of M , is a multiple of the Fisher–Rao metric. The situation for a 1-dimensional manifold is described at the end of the paper. Our result holds for smooth positive probability densities on a compact manifold. However, the proof can be adapted to a suitable (and there are many choices) space of densities on a non-compact manifold. Acknowledgments. This question was brought to our attention during a workshop at Öli-Hütte above Bad Gastein in Austria, July 14–20, 2014. We thank all the participants of the workshop for the friendly atmosphere and helpful discussions. Date: May 10, 2016 . 2010 Mathematics Subject Classification. Primary 58B20, 58D15.