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

Let M_μ be the set of all probability densities equivalent to a given reference probability measure μ. This set is thought of as the maximal regular (i.e., with strictly positive densities) μ-dominated statistical model. For each f ∈ M_μ we define (1) a Banach space Lf with unit ball Vf and (2) a mapping sf from a subset Uf of M_μ onto Vf, in such a way that the system (sf, Uf, f ∈ M_μ) is an affine atlas on M_μ. Moreover each parametric exponential model dominated by μ is a finite-dimensional affine submanifold and each parametric statistical model dominated by μ 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.