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. 
Martin Bauer, Martins Bruveris, Peter W. Michor
math.DG math.PR