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

Abstract

Let $(X,calX,\mu )$ be a measure space, and let $calM(X,calX,\mu )$ denote the set of the $\mu$-almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on $calM(X,calX,\mu )$ can be realized by an affine atlas whose charts are defined locally by the mappings $calM(X,calX,\mu )\supset {calU}_p\ni q\mapsto log(q/p)+K(p,q)\in B_p$, where ${calU}_p$ is a suitable open set containing $p$, $K(p,q)$ is the Kullback--Leibler relative information and $B_p$ is the vector space of centred and exponentially $(p\cdot \mu )$-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.