Tsallis and Rényi deformations linked via a new λ-duality
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Tsallis and Rényi entropies, which are monotone transformations of such other, generalize the classical Shannon entropy and the exponential family of probability distributions to non-extensive statistical physics, information theory, and statistics. The q-exponential family, as a deformed exponential family with subtractive normalization, nevertheless reflects the classical Legendre duality of convex functions as well as the associated concept of Bregman divergence. In this paper we show that a generalized λ-duality, where λ = 1 - q is the constant information-geometric curvature, induces a deformed exponential family with divisive normalization and links to Rényi entropy and optimal transport. Our λ-duality unifies the two deformation models, which differ by a mere reparameterization, and provides an elegant and deep framework to study the underlying mathematical structure. Using this duality, under which the Rényi entropy and divergence appear naturally, the λ-exponential family satisfies properties that parallel and generalize those of the exponential family. In particular, we give a new proof of the Tsallis entropy maximizing property of the q-exponential family. We also introduce a λ-mixture family which may be regared as the dual of the λ-exponential family. Finally, we discuss a duality between the λ-exponential family and the logarithmic divergence, and study its statistical consequences.
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