联合分布的信息不等式及其解释与应用
Information Inequalities for Joint Distributions, with Interpretations and Applications
AI总结：
本文基于任意子集联合熵给出联合熵的上下界，推广香农链式法则等经典结果，并揭示其与次模函数、组合数学、矩阵理论、假设检验及多学科的联系。
摘要：
针对一组随机变量的联合熵，本文在任意子集联合熵集合的基础上得到了上界和下界。这些不等式推广了香农的熵链式法则以及韩、藤重和希勒的不等式。本文建立了联合熵上界与下界之间的对偶关系。所有结果都被证明是次模函数的一般性新结果的特殊情形——因此，所给出的不等式构成了一类结构丰富的香农型不等式。这些新不等式被应用于组合数学中，获得了新的结果，例如任意图中独立集数量的界和零差错信源信道码的数量，以及矩阵理论中新的行列式不等式。本文还推导了一个新的相对熵不等式，并给出了其在假设检验方面的解释。最后，本文探讨了这些结果与经济学、计算机科学和物理学文献之间的揭示性联系。
英文摘要：
Upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies. These inequalities generalize Shannon's chain rule for entropy as well as inequalities of Han, Fujishige and Shearer. A duality between the upper and lower bounds for joint entropy is developed. All of these results are shown to be special cases of general, new results for submodular functions-- thus, the inequalities presented constitute a richly structured class of Shannon-type inequalities. The new inequalities are applied to obtain new results in combinatorics, such as bounds on the number of independent sets in an arbitrary graph and the number of zero-error source-channel codes, as well as new determinantal inequalities in matrix theory. A new inequality for relative entropies is also developed, along with interpretations in terms of hypothesis testing. Finally, revealing connections of the results to literature in economics, computer science, and physics are explored.
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。
