  The partial information decomposition (PID) aims to quantify the amount of
redundant information that a set of sources provides about a target. Here, we
show that this goal can be formulated as a type of information bottleneck (IB)
problem, termed the "redundancy bottleneck" (RB). The RB formalizes a tradeoff
between prediction and compression: it extracts information from the sources
that best predict the target, without revealing which source provided the
information. It can be understood as a generalization of "Blackwell
redundancy", which we previously proposed as a principled measure of PID
redundancy. The "RB curve" quantifies the prediction--compression tradeoff at
multiple scales. This curve can also be quantified for individual sources,
allowing subsets of redundant sources to be identified without combinatorial
optimization. We provide an efficient iterative algorithm for computing the RB
curve.
