The Exponential Statistical Manifold: Mean Parameters, Orthogonality and Space Transformations

Abstract

Let ( X,cal X,μ) be a measure space, and let cal M (X,cal X,μ) denote the set of the μ -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on cal M (X,cal X,μ) can be realized by an affine atlas whose charts are defined locally by the mappings cal M (X,cal X,μ)⊃cal U p∋q↦log(q/p)+K(p,q)∈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 centred and exponentially ( p⋅μ) -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.