Lifting statistical structures

TL;DR

This paper develops a method to lift geometric structures from statistical manifolds to higher tangent bundles, resulting in pseudo-Riemannian metrics and preserved potential functions, with applications to Lie groupoids.

Contribution

It introduces a novel lifting procedure for statistical structures to higher tangent bundles and Lie algebroids, preserving key properties and expanding geometric tools for statistical models.

Findings

Lifted structures form statistical manifolds on higher tangent bundles.

Potentials are preserved as divergence functions under lifting.

Explicit examples demonstrate applications to statistical models and Lie groupoids.

Abstract

We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids.