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
$\lambda $
-duality, where
$\lambda = 1 - q$
is to be interpreted as 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
$\lambda $
-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
$\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.