We quantify the average amount of redundant information that is transferred from a subset of relevant random source processes to a target process. To identify the relevant source processes, we consider those that are connected to the target process and in addition share a certain proportion of the total information causally provided to the target. Even if the relevant processes have no directed information exchange between them, they can still causally provide redundant information to the target. This makes it difficult to identify the relevant processes. To solve this issue, we propose the existence of a hidden redundancy process that governs the shared information among the relevant processes. We bound the redundancy by the minimal average directed redundancy from the relevant processes to the target, from the hidden redundancy process to the target, and from the hidden redundancy process to the relevant processes.

翻译：我们量化了从一组相关的随机源过程到目标过程中传输的平均冗余信息量。为识别相关源过程，我们考虑那些与目标过程相连，并且额外共享因果提供给目标的总信息中特定比例的源过程。即使这些相关过程之间不存在有向信息交换，它们仍可能因果地向目标提供冗余信息，这加大了识别相关过程的难度。为解决此问题，我们提出存在一个隐冗余过程，它支配着相关过程间的共享信息。我们将冗余界定为：从相关过程到目标的最小平均有向冗余、从隐冗余过程到目标的最小平均有向冗余，以及从隐冗余过程到相关过程的最小平均有向冗余。