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Tsallis and Rényi deformations linked via a new $λ$-duality

Tsallis and Rényi entropies, which are monotone transformations of each other, are deformations of the celebrated Shannon entropy. Maximization of these deformed entropies, under suitable constraints, leads to the $q$-exponential family which has applications in non-extensive statistical physics, information theory and statistics. In previous information-geometric studies, the $q$-exponential family was analyzed using classical convex duality and Bregman divergence. In this paper, we show that a generalized $λ$-duality, where $λ= 1 - q$ is the constant information-geometric curvature, leads to a generalized exponential family which is essentially equivalent to the $q$-exponential family and has deep connections with Rényi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $λ$-exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the Rényi entropy and divergence arise naturally, and we give a new proof of the Tsallis/Rényi entropy maximizing property of the $q$-exponential family. We also introduce a $λ$-mixture family which may be regarded as the dual of the $λ$-exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $λ$-exponential family and the $λ$-logarithmic divergence, and study its statistical consequences.