# Lie groupoids in information geometry

## TL;DR

This paper introduces a novel framework using Lie groupoids and Lie algebroids to unify and generalize contrast functions in information geometry, leading to new geometric insights and examples.

## Contribution

It establishes Lie groupoids and Lie algebroids as the proper setting for contrast functions, connecting them to metrics and connections in a unified geometric framework.

## Key findings

- Contrast functions induce pseudo-Riemannian metrics on Lie algebroids.
- The framework includes classical two-point functions and their geometric structures.
- An example of the Fubini-Study metric on Hilbert space is derived.

## Abstract

We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. If the two-form is non-degenerate, it defines a `pseudo-Riemannian' metric on the Lie algebroid and a family of Lie algebroid torsion-free connections, including the Levi-Civita connection of the metric. In this framework, the two-point functions are just functions on the pair groupoid $M\ti M$ with the `standard' metric and affine connection on the Lie algebroid $\sT M$. We study also reductions of such systems and infinite-dimensional examples. In particular, we find a contrast function defining the Fubini-Study metric on the Hilbert projective space.