A study of Rényi entropy based on the information geometry formalism

In the framework of information geometry, we explore the structure of statistical manifold Sn related to the Rényi entropy. Starting from a new family of generalized exponential distributions given by the equilibrium distribution of a canonical system described by the Rényi entropy, we introduce on Sn a dually-flat geometry endowed by a Hessian structure derived from a generalization of Fisher metric and a flat affine connection. This geometrical structure admits a Legendre transform that links the thermodynamic potentials of the underlying statistical mechanics to the dual potentials of the corresponding information geometry. A canonical divergence function à la Bregman is naturally derived in this framework. Further geometric structures, derived from other contrast functions strictly related to the Rényi entropy are explored and their link with the ?*-geometry of Amari is highlighted.