The Hilbert Riemannian structure of equivalent Gaussian measures associated with the Fisher information (Q1804703)

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 dissimilarity between two probability measures [cf., for example, \textit{S.-I. Amari}, \textit{O. E. Barndorff-Nielsen}, \textit{R. E. Kass}, \textit{S. L. Lauritzen} and \textit{C. R. Rao}, ``Differential geometry in statistical inference'' (1987; Zbl 0694.62001)]. \textit{S.-I. Amari} [``Differential- geometrical methods in statistics'' (1985; Zbl 0559.62001)] introduced a pair of dual affine connections with respect to the metric and discussed 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.