The exponential statistical manifold: mean parameters, orthogonality and space transformations.

Let $(X, Cal X, mu)$ be a measure space, and let $Cal M(X,Cal X,mu)$ denote the set of the $mu$-almost surely strictly positive probability densities. It was shown by G. Pistone and C. Sempi (1995) that the global geometry on $Cal M(X,Cal X,mu)$ can be realized by an affine atlas whose charts are defined locally by the mappings $Cal M(X,Cal X,mu)supset Cal U_p i q mapsto log(q/p) + K(p,q)in B_p$, where $Cal U_p$ is a suitable open set containing $p$, $K(p,q)$ is the Kullback-Leibler relative information and $B_p$ is the vector space of centered and exponentially $(pcdotmu)$-integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, that is measurable transformation of the sample space. A generalization of the mixed parameterization method for exponential models is also presented.