Entropy and relative or cross entropy measures are two very fundamental concepts in information theory and are also widely used for statistical inference across disciplines. The related optimization problems, in particular the maximization of the entropy and the minimization of the cross entropy or relative entropy (divergence), are essential for general logical inference in our physical world. In this paper, we discuss a two parameter generalization of the popular R\'{e}nyi entropy and associated optimization problems. We derive the desired entropic characteristics of the new generalized entropy measure including its positivity, expandability, extensivity and generalized (sub-)additivity. More importantly, when considered over the class of sub-probabilities, our new family turns out to be scale-invariant. We also propose the corresponding cross entropy and relative entropy measures and discuss their geometric properties including generalized Pythagorean results over $\beta$-convex sets. The maximization of the new entropy and the minimization of the corresponding cross or relative entropy measures are carried out explicitly under the non-extensive (`third-choice') constraints given by the Tsallis' normalized $q$ expectations which also correspond to the $\beta$-linear family of probability distributions. Important properties of the associated forward and reverse projection rules are discussed along with their existence and uniqueness. In this context, we have come up with, for the first time, a class of entropy measures -- a subfamily of our two-parameter generalization -- that leads to the classical (extensive) exponential family of MaxEnt distributions under the non-extensive constraints. Other members of the new entropy family, however, lead to the power-law type generalized $q$-exponential MaxEnt distributions which is in conformity with Tsallis' nonextensive theory.
翻译:信息理论中有两个非常基本的概念, 以及相对的或交叉的度量, 它们是信息理论中两个非常基本的概念, 并且被广泛用于跨学科的统计推导。 相关的优化问题, 特别是最小化和最小化的交叉环流或相对的环流( 振动), 是我们物理世界中一般逻辑推导的关键。 在本文中, 我们讨论流行的 R\ { e} 尼向量和相关的优化问题的两个参数的概括性。 我们从新通用的度量度测量中得出理想的进量特征, 包括它的正态、 扩张性、 扩展性以及普遍( 次向量) 。 当我们考虑分量的分量值时, 我们的新家系的分量性分量分量值会变成最小化的分量值, 其它直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系直系。