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Legendre duality: from thermodynamics to information geometry

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Abstract

This paper reviews the role of convex duality in Information Geometry. It clarifies the notion of bi-orthogonal coordinates associated with Legendre duality by treating its two underlying aspects separately: as a dual coordinate system and as a bi-orthogonal frame. It addresses the deformation of exponential families in a way that stills preserves the dually-flat geometry of 1- and (-1)-connections. The deformation involves a metric which generalizes the Fisher–Rao metric controlled by one degree of freedom and a pair of connections controlled by an additional degree of freedom.

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Authors and Affiliations

  1. Department of Physics, Universiteit Antwerpen, Universiteitsplein 1, Wilrijk, Antwerpen, 2610, Belgium

    Jan Naudts

  2. Department of Psychology and Department of Statistics, University of Michigan, 530 Church Street, Ann Arbor, Michigan, 48109, USA

    Jun Zhang

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  1. Jan Naudts
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Correspondence to Jan Naudts.

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Jun Zhang is a Co-Editor of the journal. Jan Naudts is a board member of the journal. Both were not involved in the peer review or handling of the manuscript. On behalf of all authors, the corresponding author states that there is no other potential conflict of interest to declare.

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Communicated by Hiroshi Matsuzoe.

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Naudts, J., Zhang, J. Legendre duality: from thermodynamics to information geometry. Info. Geo. 7 (Suppl 1), 623–649 (2024). https://doi.org/10.1007/s41884-023-00121-0

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