R. G. James, J. Emenheiser, and J. P. Crutchfield

ABSTRACT: The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable X_{i} has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X_{0} and X_{1} redundantly share with Y, what X_{0} uniquely shares with Y, what X_{1} uniquely shares with Y, and finally what X_{0} and X_{1} synergistically share with Y. Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique informations. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PID's meaning—interpretations not present in PID's definition but that, we argue, need to be explicit. These reveal an inconsistency between thirdorder connected information, twoway secret key agreement rate, and synergy. Similar difficulties arise with a popular PID measure in light the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.