  Partial information decompositions (PIDs), which quantify information
interactions between three or more variables in terms of uniqueness, redundancy
and synergy, are gaining traction in many application domains. However, our
understanding of the operational interpretations of PIDs is still incomplete
for many popular PID definitions. In this paper, we discuss the operational
interpretations of unique information through the lens of two well-known PID
definitions. We reexamine an interpretation from statistical decision theory
showing how unique information upper bounds the risk in a decision problem. We
then explore a new connection between the two PIDs, which allows us to develop
an informal but appealing interpretation, and generalize the PID definitions
using a common Lagrangian formulation. Finally, we provide a new PID definition
that is able to capture the information that is unique. We also show that it
has a straightforward interpretation and examine its properties.
