Mutual information, synergy and some curious phenomena for simple channels

Suppose we are allowed to observe two equally noisy versions of some signal X, where the level of the noise is fixed. We are given a choice: we can either observe two independent noisy versions of X, or two correlated ones. We show that, contrary to what classical statistical intuition suggests, it is often the case that correlated data is more valuable than independent data. We investigate this phenomenon in a variety of contexts, we give numerous examples for standard families of channels, and we present general sufficient conditions for deciding this dilemma. One of these conditions draws an interesting connection with the information-theoretic notion of "synergy," which has received a lot of attention in the neuroscience literature recently