  In this paper, we define a new measure of the redundancy of information from
a fault tolerance perspective. The partial information decomposition (PID)
emerged last decade as a framework for decomposing the multi-source mutual
information $I(T;X_1, ..., X_n)$ into atoms of redundant, synergistic, and
unique information. It built upon the notion of redundancy/synergy from
McGill's interaction information (McGill 1954). Separately, the redundancy of
system components has served as a principle of fault tolerant engineering, for
sensing, routing, and control applications. Here, redundancy is understood as
the level of duplication necessary for the fault tolerant performance of a
system. With these two perspectives in mind, we propose a new PID-based measure
of redundancy $I_{\text{ft}}$, based upon the presupposition that redundant
information is robust to individual source failures. We demonstrate that this
new measure satisfies the common PID axioms from (Williams 2010). In order to
do so, we establish an order-reversing correspondence between collections of
source-fallible instantiations of a system, on the one hand, and the PID
lattice from (Williams 2010), on the other.
