Dual Connections in Nonparametric Classical Information Geometry

TL;DR

This paper develops an infinite-dimensional information geometry framework using exponential Orlicz spaces, introducing dual connections, mixture extensions, and lpha-derivatives, advancing the mathematical understanding of probability density manifolds.

Contribution

It constructs a novel infinite-dimensional information manifold, characterizes mixture extensions, and defines dual connections and lpha-derivatives in this setting.

Findings

Convex mixtures of densities lie on the same connected component.

Mixture and exponential parallel transports are dual with respect to Fisher information.

lpha-derivatives are convex mixtures of extremal derivatives.

Abstract

We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixture can be extended to an open segment containing the extreme points. For this class, we define an infinite-dimensional analogue of the mixture parallel transport and prove that it is dual to the exponential parallel transport with respect to the Fisher information. We also define {\alpha}-derivatives and prove that they are convex mixtures of the extremal (\pm 1)-derivatives.