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.

Decomposing and Tracing Mutual Information by Quantifying Reachable Decision Regions.

The idea of a partial information decomposition (PID) gained significant attention for attributing the components of mutual information from multiple variables about a target to being unique, redundant/shared or synergetic. Since the original measure for this analysis was criticized, several alternatives have been proposed but have failed to satisfy the desired axioms, an inclusion-exclusion principle or have resulted in negative partial information components. For constructing a measure, we interpret the achievable type I/II error pairs for predicting each state of a target variable (reachable decision regions) as notions of pointwise uncertainty. For this representation of uncertainty, we construct a distributive lattice with mutual information as consistent valuation and obtain an algebra for the constructed measure. The resulting definition satisfies the original axioms, an inclusion-exclusion principle and provides a non-negative decomposition for an arbitrary number of variables. We demonstrate practical applications of this approach by tracing the flow of information through Markov chains. This can be used to model and analyze the flow of information in communication networks or data processing systems.