Summary: 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 \({\mathcal M}_f\) 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 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 [C. R. Rao, Sankyā 9, 246-248 (1949)] and the Hilbert bundle introduced by Amari [S.-I. Amari et al., Differential geometry in statistical inference. (1987; Zbl 0694.62001)].