Let be a measure space, and let denote the set of the -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on can be realized by an affine atlas whose charts are defined locally by the mappings , where is a suitable open set containing , is the Kullback--Leibler relative information and is the vector space of centred and exponentially -integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.
"The exponential statistical manifold: mean parameters, orthogonality and space transformations." Bernoulli 5 (4) 721 - 760, august 1999.