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

Abstract

Let $\mathscr{M}_\mu$ 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 $f \in \mathscr{M}_\mu$ we define (1) a Banach space $L_f$ with unit ball $\mathscr{V}_f$ and (2) a mapping $s_f$ from a subset $\mathscr{U}_f$ of $\mathscr{M}_\mu$ onto $\mathscr{V}_f$, in such a way that the system $(s_f, \mathscr{U}_f, f \in \mathscr{M}_\mu)$ is an affine atlas on $\mathscr{M}_\mu$. 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.