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A study of Rényi entropy based on the information geometry formalism

In the framework of information geometry, we explore the structure of statistical manifold 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 a dually-flat geometry endowed by a Hessian structure derived from a generalization of the 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.