Tsallis and R\'{e}nyi deformations linked via a new $\lambda$-duality

Tsallis and R\'{e}nyi 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 $\lambda$-duality, where $\lambda = 1 - q$ 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 R\'{e}nyi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $\lambda$-exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the R\'{e}nyi entropy and divergence arise naturally, and we give a new proof of the Tsallis/R\'{e}nyi entropy maximizing property of the $q$-exponential family. We also introduce a $\lambda$-mixture family which may be regarded as the dual of the $\lambda$-exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $\lambda$-exponential family and the $\lambda$-logarithmic divergence, and study its statistical consequences.