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Tsallis and R\'{e}nyi entropies, which are monotone transformations of eachother, are deformations of the celebrated Shannon entropy. Maximization ofthese deformed entropies, under suitable constraints, leads to the$q$-exponential family which has applications in non-extensive statisticalphysics, information theory and statistics. In previous information-geometricstudies, the $q$-exponential family was analyzed using classical convex dualityand Bregman divergence. In this paper, we show that a generalized$\lambda$-duality, where $\lambda = 1 - q$ is the constantinformation-geometric curvature, leads to a generalized exponential familywhich is essentially equivalent to the $q$-exponential family and has deepconnections with R\'{e}nyi entropy and optimal transport. Using thisgeneralized convex duality and its associated logarithmic divergence, we showthat our $\lambda$-exponential family satisfies properties that parallel andgeneralize those of the exponential family. Under our framework, the R\'{e}nyientropy and divergence arise naturally, and we give a new proof of theTsallis/R\'{e}nyi entropy maximizing property of the $q$-exponential family. Wealso introduce a $\lambda$-mixture family which may be regarded as the dual ofthe $\lambda$-exponential family, and connect it with other mixture-typefamilies. Finally, we discuss a duality between the $\lambda$-exponentialfamily and the $\lambda$-logarithmic divergence, and study its statisticalconsequences.

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