The partial information decomposition (PID) framework is concerned with decomposing the information that a set of random variables has with respect to a target variable into three types of components: redundant, synergistic, and unique. Classical information theory alone does not provide a unique way to decompose information in this manner and additional assumptions have to be made. Recently, Kolchinsky proposed a new general axiomatic approach to obtain measures of redundant information, based on choosing an order relation between information sources (equivalently, order between communication channels). In this paper, we exploit this approach to introduce three new measures of redundant information (and the resulting decompositions) based on well-known preorders between channels, thus contributing to the enrichment of the PID landscape. We relate the new decompositions to existing ones, study some of their properties, and provide examples illustrating their novelty. As a side result, we prove that any preorder that satisfies Kolchinsky's axioms yields a decomposition that meets the axioms originally introduced by Williams and Beer when they first propose the PID.

翻译：部分信息分解（PID）框架旨在将一组随机变量关于目标变量的信息分解为三种成分：冗余信息、协同信息和独特信息。经典信息理论本身并未提供此种分解的唯一方式，因此需要引入额外假设。最近，Kolchinsky提出了一种新的通用公理化方法，通过选择信息源之间的序关系（等价于通信信道间的序关系）来度量冗余信息。本文基于该方法，利用信道间几种经典预序关系，引入了三种新的冗余信息度量（及其对应的分解），从而丰富了PID的理论体系。我们将新分解与现有分解进行关联，研究其部分性质，并通过算例说明其新颖性。作为附带结果，我们证明了任何满足Kolchinsky公理的预序关系所产生的分解，均符合Williams和Beer首次提出PID时所建立的基本公理体系。