Geometry Induced by a Generalization of Rényi Divergence
نویسندگان
چکیده
In this paper, we propose a generalization of Rényi divergence, and then we investigate its induced geometry. This generalization is given in terms of a φ-function, the same function that is used in the definition of non-parametric φ-families. The properties of φ-functions proved to be crucial in the generalization of Rényi divergence. Assuming appropriate conditions, we verify that the generalized Rényi divergence reduces, in a limiting case, to the φ-divergence. In generalized statistical manifold, the φ-divergence induces a pair of dual connections D(−1) and D(1). We show that the family of connections D(α) induced by the generalization of Rényi divergence satisfies the relation D(α) = 1−α 2 D (−1) + 1+α 2 D (1), with α ∈ [−1, 1].
منابع مشابه
A Preferred Definition of Conditional Rényi Entropy
The Rényi entropy is a generalization of Shannon entropy to a one-parameter family of entropies. Tsallis entropy too is a generalization of Shannon entropy. The measure for Tsallis entropy is non-logarithmic. After the introduction of Shannon entropy , the conditional Shannon entropy was derived and its properties became known. Also, for Tsallis entropy, the conditional entropy was introduced a...
متن کاملOn The Equivalence of Projections In Relative $\alpha$-Entropy and R\'enyi Divergence
The aim of this work is to establish that two recently published projection theorems, one dealing with a parametric generalization of relative entropy and another dealing with Rényi divergence, are equivalent under a correspondence on the space of probability measures. Further, we demonstrate that the associated “Pythagorean” theorems are equivalent under this correspondence. Finally, we apply ...
متن کاملMinimization Problems Based on a Parametric Family of Relative Entropies I: Forward Projection
Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative α-entropies (denoted Iα), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual r...
متن کاملWyner's Common Information under Rényi Divergence Measures
We study a generalized version of Wyner’s common information problem (also coined the distributed sources simulation problem). The original common information problem consists in understanding the minimum rate of the common input to independent processors to generate an approximation of a joint distribution when the distance measure used to quantify the discrepancy between the synthesized and t...
متن کاملA duality relation connecting different quantum generalizations of the conditional Rényi entropy
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. This generalizes the well-known duality relation...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Entropy
دوره 18 شماره
صفحات -
تاریخ انتشار 2016