关于Mathai熵测度的一则注记
A Note on Mathai's Entropy Measure
AI总结：
该注记梳理了熵的三类划分，论证Mathai熵测度在不同类型变量下形式一致，可直接用变分法在类矩条件下优化得到各类分布，无需Tsallis熵所需的中间escort分布，并举例说明其可导出的分布类型。
摘要：
在文献[8]中，作者将熵划分为三类：作为热力学量的熵、作为信息产生度量的熵，以及作为统计推断手段的熵。Mathai提出的一种熵测度属于第二类和第三类。研究表明，无论涉及的变量是实值还是复值的标量、向量或矩阵变量，该熵测度的形式均保持一致。若在类矩条件下对该熵测度进行优化，可得到适用于不同领域的各类密度函数。与Tsallis熵（文献[9]）不同，它无需借助中间的escort分布即可得到预期结果，可直接应用变分法在Mathai熵框架下推导目标结果。作为非广延统计力学领域基础的Tsallis熵，是文献[7]中所研究的Havrda-Charvat的α广义熵的一种修正形式。本文还举例说明了通过优化Mathai熵可得到的各类分布。
英文摘要：
In a paper [8] the authors classify entropy into three categories, as a thermodynamics quantity, as a measure of information production, and as a means of statistical inference. An entropy measure introduced by Mathai falls into the second and third categories. It is shown that this entropy measure is the same whether the variables involved are real or complex scalar, vector, or matrix variables. If the entropy measure is optimized under some moment-like conditions then one can obtain various types of densities which are applicable in different areas. Unlike Tsallis' entropy [9], it does not need an intermediary escort distribution to yield the desired results. Calculus of variation can be directly applied to obtain the desired results under Mathai's entropy. Tsallis' entropy, which is the basis of the area of non-extensive statistical mechanics, is a modified version of the $α$-generalized entropy of Havrda-Charvat considered in [7]. Various types of distributions that can be obtained through optimization of Mathai's entropy, are illustrated in this paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
