  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.
