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.
翻译:Tsallis 和 R\ { e} ny exproditions, 它们是彼此的单质变异, 是庆祝的香农变形的变形。 这些变形的异种, 在适当的限制下, 使这些变形的异种变形的变异性变异性变异性变异性变异性变异性变异性变异性。 这些变异性变异性在适当的限制下, 导致美元- Expensional- Expendial- family, 基本上相当于美元- Expensive- 统计学、 信息理论理论和统计学。 使用这种典型的共性变异性变性变性, 我们展示了我们的美元- 美元- 美元- 美元- 美元- 质量, 其中美元= qqureental- labal- laum 家庭变异性变性变性变性变性关系。