Non-Negative Decomposition of Multivariate Information: From Minimum to Blackwell-Specific Information

Partial information decompositions (PIDs) aim to categorize how a set of source variables provides information about a target variable redundantly, uniquely, or synergetically. The original proposal for such an analysis used a lattice-based approach and gained significant attention. However, finding a suitable underlying decomposition measure is still an open research question at an arbitrary number of discrete random variables. This work proposes a solution with a non-negative PID that satisfies an inclusion-exclusion relation for any f-information measure. The decomposition is constructed from a pointwise perspective of the target variable to take advantage of the equivalence between the Blackwell and zonogon order in this setting. Zonogons are the Neyman-Pearson region for an indicator variable of each target state, and f-information is the expected value of quantifying its boundary. We prove that the proposed decomposition satisfies the desired axioms and guarantees non-negative partial information results. Moreover, we demonstrate how the obtained decomposition can be transformed between different decomposition lattices and that it directly provides a non-negative decomposition of Rényi-information at a transformed inclusion-exclusion relation. Finally, we highlight that the decomposition behaves differently depending on the information measure used and how it can be used for tracing partial information flows through Markov chains.