Nonnegative Decomposition of Multivariate Information (1004.2515v1)

Abstract: Of the various attempts to generalize information theory to multiple variables, the most widely utilized, interaction information, suffers from the problem that it is sometimes negative. Here we reconsider from first principles the general structure of the information that a set of sources provides about a given variable. We begin with a new definition of redundancy as the minimum information that any source provides about each possible outcome of the variable, averaged over all possible outcomes. We then show how this measure of redundancy induces a lattice over sets of sources that clarifies the general structure of multivariate information. Finally, we use this redundancy lattice to propose a definition of partial information atoms that exhaustively decompose the Shannon information in a multivariate system in terms of the redundancy between synergies of subsets of the sources. Unlike interaction information, the atoms of our partial information decomposition are never negative and always support a clear interpretation as informational quantities. Our analysis also demonstrates how the negativity of interaction information can be explained by its confounding of redundancy and synergy.

• The paper introduces a nonnegative decomposition framework that resolves the issue of negative interaction information by redefining redundancy.
• It constructs a redundancy lattice to break down multivariate interactions into nonnegative partial information atoms for clearer interpretation.
• The framework offers practical insights for analyzing complex systems in neuroscience, genetics, and systems biology research.

Nonnegative Decomposition of Multivariate Information

The paper "Nonnegative Decomposition of Multivariate Information" by Paul L. Williams and Randall D. Beer provides a novel approach to resolving certain limitations of traditional multivariate information measures, particularly focusing on interaction information. The authors propose a theoretical framework that redefines redundancy and introduces a systematic decomposition of information across multiple variables. This paper contributes significantly to the understanding and application of information theory in complex systems where multivariate interactions are critical.

Background and Motivation

The analysis of multivariate interactions traditionally utilizes measures like total correlation and interaction information. However, these measures have limitations; notably, interaction information can sometimes result in negative values, which complicates its interpretation as a measure of the synergy or redundancy among variables. This paper aims to address these interpretational issues by proposing a reformulation grounded in nonnegative decompositions.

Redefinition of Redundancy and Lattice Structure

The authors begin by redefining redundancy as the minimum information any source provides about each possible outcome of a variable, averaged over all outcomes. This notion of redundancy supports the construction of a redundancy lattice, a structure that represents how different subsets of variables contribute redundantly to the information about a target variable. The redundancy lattice helps classify interactions between variables more effectively compared to previous methods.

Partial Information Decomposition

A cornerstone of this paper is the introduction of partial information decomposition. Unlike interaction information, which can mix synergy and redundancy, partial information decomposition delineates contributions of each variable subset into nonnegative partial information atoms. This ensures that every piece of information has a clear interpretative path within the system of variables, resulting in a comprehensive breakdown of the information dynamics.

Practical and Theoretical Implications

The implications of this work are both practical and theoretical:

1. Theoretical Clarity: By ensuring that all terms in the decomposition are nonnegative, the framework offers a more intuitive understanding of multivariate interactions. This clarity will aid researchers in diverse fields such as neuroscience, genetics, and systems biology, where understanding the precise structure of variable interactions is crucial.
2. Analytical Utility: The proposed decomposition offers a robust tool for the analysis of systems with multiple interacting components. The redundancy lattice, in particular, presents a structured way to navigate the complex landscape of variable interactions, opening new directions for research in the quantification of informational synergy and redundancy.

Future Directions

This framework sets the stage for several lines of inquiry in information theory and its applications. Future work could involve exploring efficient computational methods to handle the lattice structure for high-dimensional datasets, given that the number of potential interactions grows exponentially with the number of variables.

Additionally, applying this framework in empirical studies, notably those in neuroscience and bioinformatics, could validate its utility in practical scenarios. The clear differentiation between redundancy and synergy could lead to more precise models of information processing in biological and artificial systems.

In summary, this paper positions itself as an important refinement in multivariate information analysis, offering a comprehensive and nonnegative approach to understanding complex interactions. Such developments not only equip researchers with a refined analytical tool but also enhance the theoretical foundations of information theory in the context of multivariable systems.