An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one

Let M(mu) be the set of all probability densities equivalent to a given reference probability measure mu. This set is thought of as the maximal regular (i.e., with strictly positive densities) mu-dominated statistical model. For each f is an element of M(mu) We define (1) a Banach space L(f) with unit ball V-f and(2) a mapping sf from a subset U-f of M(mu) onto V-f, in such a way that the system (s(f), U-f, f is an element of M(mu)) is an affine atlas on M(mu). Moreover each parametric exponential model dominated by mu is a finite-dimensional affine submanifold and each parametric statistical model dominated by mu with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.