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Bivariate Partial Information Decomposition (PID) describes how the mutualinformation between a random variable M and two random variables Y and Z isdecomposed into unique, redundant, and synergistic terms. Recently, PID hasshown promise as an emerging tool to understand biological systems and biasesin machine learning. However, computing PID is a challenging problem as ittypically involves optimizing over distributions. In this work, we study theproblem of computing PID in two systems: the Poisson system inspired by the'ideal Poisson channel' and the multinomial system inspired by multinomialthinning, for a scalar M. We provide sufficient conditions for both systemsunder which closed-form expressions for many operationally-motivated PID can beobtained, thereby allowing us to easily compute PID for these systems. Ourproof consists of showing that one of the unique information terms is zero,which allows the remaining unique, redundant, and synergistic terms to beeasily computed using only the marginal and the joint mutual information.

Computing Unique Information for Poisson and Multinomial Systems