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