Affine statistical bundle modeled on a Gaussian Orlicz–Sobolev space

Abstract

The dually flat structure of statistical manifolds can be derived in a non-parametric way from a particular case of affine space defined on a qualified set of probability measures. The statistically natural displacement mapping of the affine space depends on the notion of Fisher’s score. The model space must be carefully defined if the state space is not finite. Among various options, we discuss how to use Orlicz–Sobolev spaces with Gaussian weight. Such a fully non-parametric set-up provides tools to discuss intrinsically infinite-dimensional evolution problems