Lie groupoids in information geometry

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.

arXiv comment19 pages

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• Nonassociative analogs of Lie groupoids
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• Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
• Discrete mechanics on unitary octonions
• G-dual teleparallel connections in Information Geometry
• On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality
• Geometry from divergence functions and complex structures
• Statistical Geometry and Hessian Structures on Pre-Leibniz Algebroids