The Hilbert Riemannian structure of equivalent Gaussian measures associated with the Fisher information

Abstract

Rao has firstly introduced the Riemannian structure associated with the Fisher information matrix over a finite dimensional parametrized statistical model. He proposed the Riemannian distance as a measure of dissimilality between two probability measures, (cf. [2], for example.) In [1], Amari introduced a pair of dual affine connections with respect to the metric and discussed of the differential geometry of the space of a finite dimensional parametrized statistical model. It provides a differential geometrical meaning to statistical inference. In the present paper, we realize the above idea for a family of equivalent (i.e.,mutually absolute continuous) Gaussian measures on a Banach space. Our main result is as follows. Let 5 b e a real separable Banach space and P be a centered gaussian measure on the topological dual of B (cf. [6]). The covariance of P naturally determines the Hubert space H. i.e., for arbitrary x1 and x2eB, let