  We consider the problem of decomposing the total mutual information conveyed
by a pair of predictor random variables about a target random variable into
redundant, unique and synergistic contributions. We focus on the relationship
between "redundant information" and the more familiar information-theoretic
notions of "common information". Our main contribution is an impossibility
result. We show that for independent predictor random variables, any common
information based measure of redundancy cannot induce a nonnegative
decomposition of the total mutual information. Interestingly, this entails that
any reasonable measure of redundant information cannot be derived by
optimization over a single random variable.
