Gomes André F C, Figueiredo Mário A T
Instituto de Telecomunicações and LUMLIS (Lisbon ELLIS Unit), Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal.
Entropy (Basel). 2023 Jun 25;25(7):975. doi: 10.3390/e25070975.
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, contributing to the enrichment of the PID landscape. We relate the new decompositions to existing ones, study several 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 proposed PID.
部分信息分解（PID）框架关注的是将一组随机变量相对于目标变量所具有的信息分解为三种类型的成分：冗余信息、协同信息和唯一信息。仅经典信息论本身并不能提供以这种方式分解信息的唯一方法，必须做出额外的假设。最近，科尔钦斯基提出了一种新的通用公理方法，基于选择信息源之间的序关系（等效地，通信信道之间的序关系）来获得冗余信息的度量。在本文中，我们利用这种方法引入了基于信道之间著名的预序关系的三种新的冗余信息度量（以及由此产生的分解），这有助于丰富PID的研究领域。我们将新的分解与现有的分解联系起来，研究它们的几个性质，并提供示例来说明它们的新颖性。作为一个附带结果，我们证明了任何满足科尔钦斯基公理的预序关系都会产生一种分解，该分解符合威廉姆斯和比尔最初提出PID时引入的公理。
