An Infinite--Dimensional Geometric Structure on the Space of All Probability Measures Equivalent to a Given One

Let $\mathcal{M}_{\mu}$ 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 $f\in\mathcal{M}_{\mu}$ we define (1) a Banach space $L_f$ with unit ball $\mathcal{V}_f$ and (2) a mapping $s_f$ from a subset $\mathcal{U}_f$ of $\mathcal{M}_{\mu}$ onto $\mathcal{V}_f$, in such a way that the system $(s_f,\mathcal{U}_f,f\in\mathcal{M}_{\mu})$ is an affine atlas on $\mathcal{M}_{\mu}$ . Moreover each parametrci 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 gloabl geometric framework given by the manifold structure adds some insighty 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.