Legendre duality : from thermodynamics to information geometry

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.