An extended Čencov characterization of the information metric

Abstract

Abstract.Cencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In Cencov's theorem, the underlying differentiable manifold is the probability simplex E"x, = 1, x¡ > 0. For some purposes of using geometry to obtain insights about probability, it is more convenient to regard the simplex as a hypersurface in the positive cone. In the present paper Cencov's result is extended to the positive cone. The proof uses standard techniques of differential geometry but does not use the language of category theory.1. Introduction. There has been a good deal of interest in the use of differential geometry to interpret certain operations on probability distributions in statistics [2,4,5], biomathematics [9], thermodynamics [7], and information theory [3]. Further references are to be found in those cited above. Much of this literature begins by introducing a Riemannian metric which is generated by the Fisher information matrix. One reason for singling out this particular metric is to be found in a theorem of Cencov [4, Theorem 11.1 or Lemma 11.3], which characterizes this information metric on the probability simplex as the only metric having a certain invariance property under some probabilistically natural mappings.In this paper, we develop a characterization theorem which is closely related to Cencov's. The principal difference between the two theorems is that we characterize Riemannian metrics on the positive cones R^ = {x = (xx,..., x"): x¡ > 0}, while Cencov characterizes them on the probability Simplexes Sn_x = {x g R^:Ex, = 1}. As will be seen later, the connection between geometry and probability is enhanced if Sn_i is regarded as a surface in the differentiable manifold R*. In addition, some of Shahshahani's development [9] requires a metric on R+. A second difference from Cencov's theorem is that neither the statement of our result nor the proof use the language of category theory.

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