Statistical manifold as an affine space: A functional equation approach

A statistical manifold Mμ consists of positive functions f such that fdμ defines a probability measure. In order to define an atlas on the manifold, it is viewed as an affine space associated with a subspace of the Orlicz space LΦ. This leads to a functional equation whose solution, after imposing the linearity constrain in line with the vector space assumption, gives rise to a general form of mappings between the affine probability manifold and the vector (Orlicz) space. These results generalize the exponential statistical manifold and clarify some foundational issues in non-parametric information geometry.