Quantifying unique information (1311.2852v2)

Abstract: We propose new measures of shared information, unique information and synergistic information that can be used to decompose the multi-information of a pair of random variables $(Y,Z)$ with a third random variable $X$. Our measures are motivated by an operational idea of unique information which suggests that shared information and unique information should depend only on the pair marginal distributions of $(X,Y)$ and $(X,Z)$. Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

• The paper introduces a framework that decomposes multi-information into distinct shared, unique, and synergistic components.
• The paper employs marginal distributions of variable pairs to define invariant measures that satisfy non-negativity in information theory.
• The paper’s findings offer both theoretical insights and practical applications, notably in neuroscience, by clarifying information redundancy and synergy.

Essay on "Quantifying Unique Information"

In the paper "Quantifying Unique Information," Nils Bertschinger et al. present a theoretical framework for decomposing the multi-information of a set of random variables into distinct components: shared information, unique information, and synergistic information. This decomposition is aimed at better understanding how multiple variables convey information regarding a target variable. The primary focus is on a triplet of random variables $(X, Y, Z)$, where $X$ is the variable of interest, and $Y$ and $Z$ represent observed variables that might carry overlapping or distinct information about it.

Background and Motivation

The decomposition of mutual information into unique, shared, and synergistic components extends the classic definitions of mutual and Shannon information. Traditional methods measure the extent to which observing a particular variable reduces uncertainty about the variable of interest. However, when dealing with multiple observations, these methods do not distinguish between information that is redundant, uniquely provided by one variable, or synergistically provided only by the combination of multiple variables.

Methodological Approach

The approach leverages the theoretical construct of multi-information, particularly operationalizing the construct of unique information. The paper posits that if a variable $Y$ carries unique information about another variable $X$ with respect to a third variable $Z$, this unique information can be extracted or evidenced in specific decision-making scenarios. The authors formulate this by creating a set of properties that any reasonable measure of unique information should satisfy.

The critical assumption underpinning the authors' constructs is that the unique information $UI(X:Y\setminus Z)$ should depend solely on the marginal distributions of the pairs $(X, Y)$ and $(X, Z)$. The paper effectively exploits this assumption to define and analyze several mathematical formulations, concluding that shared and unique information obey this invariance property.

Numerical Results

The authors explore properties of their proposed measures and demonstrate non-negativity across three defined functions. These functions correspond to shared information $SI$, unique information $UI$, and synergistic information $CI$. Numerical solutions illustrate these functions' behavior, emphasizing their interpretative value in scenarios involving redundancy and synergy. Notably, the paper identifies conditions under which each type of information vanishes, lending operational clarity to their theoretical definitions.

Theoretical and Practical Implications

The implications of this research are twofold: theoretical rigor and practical applicability. Theoretically, the paper enriches the existing information-theoretic framework by articulating a structure for understanding variable interactions. Practically, these findings are highly relevant to fields such as neuroscience, where understanding how information is processed and integrated is paramount. For instance, discerning whether certain neural signals are redundant, unique, or synergistic can enhance our understanding of cognitive processes.

Future Developments and Speculations

Further examination is anticipated in broadening this approach to sets of more than three variables. The current formulation provides a bivariate decomposition and raises questions about extending this decomposition into a more expansive, multivariate domain. The approach will need modification to accommodate and unravel the complexities arising in systems where interdependencies are layered across multiple dimensions.

The essay reflects key methodological aspects of the paper while recognizing the broader implications of the proposed information decomposition framework. Researchers in information theory and related fields will find the proposed measures promising for a range of applications requiring nuanced understandings of variable interrelationships.