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.

翻译：双变量部分信息分解（PID）描述了随机变量 M 与两个随机变量 Y 和 Z 之间的互信息如何分解为独有、冗余和协同项。近年来，PID 作为一种新兴工具在理解生物系统和机器学习中的偏差方面展现出潜力。然而，计算 PID 是一个具有挑战性的问题，因为通常需要对分布进行优化。本文研究了在两种系统中计算 PID 的问题：受“理想泊松信道”启发的泊松系统和受多项稀疏化启发的多项系统（针对标量 M）。我们为这两种系统提供了充分条件，使得许多基于操作动机的 PID 可以获得闭式表达式，从而能够轻松计算这些系统的 PID。我们的证明过程表明，其中一个独有信息项为零，这使得剩余的独有、冗余和协同项仅需利用边缘互信息和联合互信息即可轻松计算。