从部分-整体关系和形式逻辑理解信息分解
Bits and Pieces: Understanding Information Decomposition from Part-whole Relationships and Formal Logic
摘要：
部分信息分解（PID）试图将一组源变量包含的关于目标变量的多元互信息分解成基本片段，即所谓的“信息原子”。每个原子都描述了一种不同的方式，其中源可以包含有关目标的信息。在本文中，我们表明，第一，整个理论的部分信息分解可以从考虑信息的贡献之间的基本parhood关系。这种处理问题的方法具有直接表征信息原子的优点，而不是通过冗余概念采取间接方法。其次，我们描述了PID和形式逻辑之间的一些有趣的联系。特别是，我们将展示如何定义一个衡量PID的基础上所提供的信息，某些语句的源实现。此外，我们展示了如何PID理论的数学格结构可以被翻译成一个同构的逻辑语句的结构与一个特别简单的顺序关系：逻辑蕴涵。从这些考虑中得出的结论是，部分信息分解有三个同构的“世界”，即数学上描述由一组源携带的关于目标的信息的分解的三种等价方式：亲子关系的世界，逻辑陈述的世界，以及反链的世界，威廉姆斯和比尔在他们最初对PID理论的阐述中使用了反链的世界。我们还展示了如何parthood的角度提供了一个系统的方法来回答一种类型的问题，已经在PID领域讨论：是否可以唯一地确定部分信息分解的基础上的概念，而不是冗余信息。
英文摘要：
Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so called "atoms of information". Each atom describes a distinct way in which the sources may contain information about the target. In this paper we show, first, that the entire theory of partial information decomposition can be derived from considerations of elementary parthood relationships between information contributions. This way of approaching the problem has the advantage of directly characterizing the atoms of information, instead of taking an indirect approach via the concept of redundancy. Secondly, we describe several intriguing links between PID and formal logic. In particular, we show how to define a measure of PID based on the information provided by certain statements about source realizations. Furthermore, we show how the mathematical lattice structure underlying PID theory can be translated into an isomorphic structure of logical statements with a particularly simple ordering relation: logical implication. The conclusion to be drawn from these considerations is that there are three isomorphic "worlds" of partial information decomposition, i.e. three equivalent ways to mathematically describe the decomposition of the information carried by a set of sources about a target: the world of parthood relationships, the world of logical statements, and the world of antichains that was utilized by Williams and Beer in their original exposition of PID theory. We additionally show how the parthood perspective provides a systematic way to answer a type of question that has been much discussed in the PID field: whether a partial information decomposition can be uniquely determined based on concepts other than redundant information.
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