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摘要翻译：
信息相关测度是多变量数据分析的有用工具，可以用来度量变量间的相关性，也可以用来描述生物和物理系统中的有序性。与信息相关的度量，如边际熵、相互/相互作用/多信息等，已被应用于许多领域，包括系统复杂性描述和生物数据分析。因此，这些度量之间的数学关系非常有趣。一般信息测度之间的关系包括格上基于M\'Obius反演的对偶关系。这些是变量集（子集按包含顺序排列）的格的对称性的直接结果。虽然这些与信息有关的度量之间的数学性质和关系是非常有趣的，但据我们所知，还没有像我们在这里所做的那样，对各种关系的全部范围进行系统的审查，也没有把这种不同的函数范围统一成一个单一的形式主义。本文基于M\\“Obius反演思想，定义了这些格上函数上的算子（M\\”Obius算子）我们证明了这些算子形成了一个与对称群S3同构的简单群。格上函数集之间的关系用算子代数透明地表示，应用于信息测度，可以导出测度之间的广泛关系。我们描述了条件对数似然和与先前定义的依赖度量之间的直接关系。代数自然是广义的，它产生了更广泛的关系。这种形式提供了信息相关测度的基本统一，但所有分配格与子集格的同构意味着这些结果的广泛潜在应用。
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英文标题：
《Multivariate information measures: a unification using M\"obius
operators on subset lattices》
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作者：
David J. Galas, Nikita A. Sakhanenko
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最新提交年份：
2016
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分类信息：
一级分类：Computer Science 计算机科学
二级分类：Information Theory 信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics 数学
二级分类：Information Theory 信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Quantitative Biology 数量生物学
二级分类：Other Quantitative Biology 其他定量生物学
分类描述：Work in quantitative biology that does not fit into the other q-bio classifications
不适合其他q-bio分类的定量生物学工作
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英文摘要：
Information related measures are useful tools for multi variable data analysis, as measures of dependence among variables, and as descriptions of order in biological and physical systems. Information related measures, like marginal entropies, mutual / interaction / multi-information, have been used in a number of fields including descriptions of systems complexity and biological data analysis. The mathematical relationships among these measures are therefore of significant interest. Relations between common information measures include the duality relations based on M\"obius inversion on lattices. These are the direct consequence of the symmetries of the lattices of the sets of variables (subsets ordered by inclusion). While the mathematical properties and relationships among these information-related measures are of significant interest, there has been, to our knowledge, no systematic examination of the full range of relationships and no unification of this diverse range of functions into a single formalism as we do here. In this paper we define operators on functions on these lattices based on the M\"obius inversion idea that map the functions into one another (M\"obius operators.) We show that these operators form a simple group isomorphic to the symmetric group S3. Relations among the set of functions on the lattice are transparently expressed in terms of the operator algebra, and, applied to the information measures, can be used to derive a wide range of relationships among measures. We describe a direct relation between sums of conditional log-likelihoods and previously defined dependency measures. The algebra is naturally generalized which yields more extensive relationships. This formalism provides a fundamental unification of information related measures, but isomorphism of all distributive lattices with the subset lattice implies broad potential application of these results.
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PDF链接：
https://arxiv.org/pdf/1601.06780
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