A Bivariate Measure of Redundant Information (1207.2080v3)

Abstract: We define a measure of redundant information based on projections in the space of probability distributions. Redundant information between random variables is information that is shared between those variables. But in contrast to mutual information, redundant information denotes information that is shared about the outcome of a third variable. Formalizing this concept, and being able to measure it, is required for the non-negative decomposition of mutual information into redundant and synergistic information. Previous attempts to formalize redundant or synergistic information struggle to capture some desired properties. We introduce a new formalism for redundant information and prove that it satisfies all the properties necessary outlined in earlier work, as well as an additional criterion that we propose to be necessary to capture redundancy. We also demonstrate the behaviour of this new measure for several examples, compare it to previous measures and apply it to the decomposition of transfer entropy.

• The paper introduces a novel bivariate measure that distinctly quantifies redundant information between variables.
• It employs information geometry and projection techniques to decompose mutual information into redundant and synergistic components.
• The method enhances analysis in fields like genetic networks and computational neuroscience by precisely isolating transfer entropy dynamics.

An Analytical Overview of Redundant Information Measurement in Probability Distributions

The paper "A Bivariate Measure of Redundant Information" by Harder, Salge, and Polani explores the intricacies of formalizing and quantifying redundant information in the field of probability distributions. This exploration is essential for the decomposition of mutual information into redundant and synergistic components—a topic that has not been comprehensively addressed by previous frameworks. In this review, we dissect the methodologies and theoretical claims presented by the authors, examining their contributions to our understanding of information dynamics among random variables.

Redundant Information and Mutual Information

Mutual information—a well-established concept in information theory—quantifies the amount of information one random variable holds about another. However, when considering multiple variables, it becomes vital to determine not only the extent of shared information but the nature of this sharing. Redundant information, in the authors' formulation, is the information that two variables individually possess about a third variable. This idea inherently demands a distinction from synergistic information, which is accessible only by observing the joint variable states, rather than each variable separately.

Formulation and Mathematical Rigor

The authors advance a new formalism that utilizes projections within the geometric landscape of probability distributions to compute redundant information. The technique hinges on a geometric interpretation—specifically, the projection of information shared between variables—and offers a more precise mechanism than traditional mutual information extensions. Importantly, the authors prove compliance with a set of prescribed axioms necessary for a functional redundancy measure, extending these axioms with a novel requirement linked to projection congruence.

Central to their method is the concept of "projective information," which provides insight into how two variables can work independently to inform us about a third variable without overlapping synergistic dynamics. The paper effectively establishes this measure by leveraging principles from information geometry, promising a more robust approach to differentiating redundant from synergistic information.

Practical and Theoretical Implications

This paper's contributions have significant repercussions on both theoretical explorations and practical applications within computational systems. For instance, the measure can be applied to decompose transfer entropy—a crucial component in understanding information flow across processes—into its independent and dependent aspects. This decomposition is instrumental in settings such as genetic regulatory networks and computational neuroscience, where pinpointing precise information transfer dynamics can enhance modeling and predictive capacity.

Additionally, the new measure holds potential for refining applications in control theory, particularly related to open-loop control mechanisms as explored by the authors. By accurately distinguishing between redundant information vibrations and state-dependent transfer, the authors' measure could provide deeper insights into system controllability dynamics.

Future Directions and Considerations

While the paper delineates a robust bivariate measure, the extension towards a multivariate framework remains a fertile ground for future research. Tackling this complex mathematical challenge could facilitate the broader application of redundancy measures in multi-variable systems—a commonplace scenario in real-world applications. Furthermore, investigating continuous random variables could advance the universality of their approach across different domains requiring nuanced information parsing.

In conclusion, this paper presents a refined, mathematically rigorous tool for identifying redundant information among variables—markedly improving upon established methodologies. As the utility of information theory continues its expansion into diverse fields, measures such as those proposed by Harder, Salge, and Polani pave the way for more profound analyses and enhanced computational capabilities.