While mutual information effectively quantifies dependence between two variables, it does not by itself reveal the complex, fine-grained interactions among variables, i.e., how multiple sources contribute redundantly, uniquely, or synergistically to a target in multivariate settings. The Partial Information Decomposition (PID) framework was introduced to address this by decomposing the mutual information between a set of source variables and a target variable into fine-grained information atoms such as redundant, unique, and synergistic components. In this work, we review the axiomatic system and desired properties of the PID framework and make three main contributions. First, we resolve the two-source PID case by providing explicit closed-form formulas for all information atoms that satisfy the full set of axioms and desirable properties. Second, we prove that for three or more sources, PID suffers from fundamental inconsistencies: we review the known three-variable counterexample where the sum of atoms exceeds the total information, and extend it to a comprehensive impossibility theorem showing that no lattice-based decomposition can be consistent for all subsets when the number of sources exceeds three. Finally, we deviate from the PID lattice approach to avoid its inconsistencies, and present explicit measures of multivariate unique and synergistic information. Our proposed measures, which rely on new systems of random variables that eliminate higher-order dependencies, satisfy key axioms such as additivity and continuity, provide a robust theoretical explanation of high-order relations, and show strong numerical performance in comprehensive experiments on the Ising model. Our findings highlight the need for a new framework for studying multivariate information decomposition.

翻译：虽然互信息能有效量化两个变量之间的依赖性，但其本身无法揭示变量间复杂而精细的交互作用，即在多元情境下多个源变量如何以冗余、独特或协同的方式共同影响目标变量。偏信息分解框架正是为解决此问题而提出，它将一组源变量与目标变量之间的互信息分解为冗余、独特及协同等精细信息原子。本文首先回顾了PID框架的公理体系与期望性质，并作出三项主要贡献。首先，我们通过给出满足全部公理与期望性质的各信息原子的显式闭型公式，完整解决了双源PID问题。其次，我们证明当源变量数量达到或超过三个时，PID存在根本性不一致：我们回顾了已知的三变量反例（其中原子信息之和超过总信息量），并将其扩展为一个完备的不可能性定理，证明当源变量超过三个时，任何基于格结构的分解都无法在所有子集上保持一致性。最后，我们摒弃PID的格结构方法以规避其不一致性，提出了多元独特信息与协同信息的显式度量。我们提出的度量基于消除高阶依赖性的新随机变量系统，满足可加性、连续性等关键公理，为高阶关系提供了稳健的理论解释，并在伊辛模型的综合实验中展现出优异的数值性能。本研究结果凸显了建立多元信息分解新框架的必要性。