Quantifying synergistic mutual information (1205.4265v6)

Abstract: Quantifying cooperation or synergy among random variables in predicting a single target random variable is an important problem in many complex systems. We review three prior information-theoretic measures of synergy and introduce a novel synergy measure defined as the difference between the whole and the union of its parts. We apply all four measures against a suite of binary circuits to demonstrate that our measure alone quantifies the intuitive concept of synergy across all examples. We show that for our measure of synergy that independent predictors can have positive redundant information.

• The paper introduces a novel measure, Synergistic Information (S₍∩∪₎), to accurately quantify the extra information when predictors combine.
• It employs a minimization strategy to separate genuine synergy from redundancy, validated through examples like XOR and AND logic gates.
• The approach enhances understanding of complex interactions in neuroscience and genetics, offering practical insights for advanced systems analysis.

Analyzing "Quantifying Synergistic Mutual Information"

The paper presented introduces a novel approach to quantifying synergistic mutual information. In the context of information theory, synergy is understood as the amount by which the whole system exceeds the union of its individual components in providing information about a target variable. This concept has significant relevance across fields such as computational biology and neuroscience, with applications ranging from genetic trait analysis to neuronal activity modeling.

Overview of Synergy Measures

Historically, the measurement of synergy has been approached through a variety of strategies, yet no consensus has emerged regarding which approach best captures the intuitive and theoretical essence of synergy. The paper critiques previous measures, namely $\operatorname{I}_{\max}$, WholeMinusSum (WMS), and Correlational Importance ($\Delta I$), illustrating their limitations through a series of examples. Most notably:

• $\mathcal{S}_{\max}$: This measure tends to overestimate synergy when unique information from predictors is misconstrued as synergistic.
• WholeMinusSum (WMS): As a signed measure, WMS struggles with redundancy, often yielding negative synergy values where redundancy is present, thereby underestimating true synergy.
• Correlational Importance ($\Delta I$): While innovative in approach, $ΔI$ sometimes exceeds mutual information bounds, indicating measurement of a fundamentally different quantity altogether.

Proposed Measure: Synergistic Information ($\mathcal{S}_{\cap\cup}$)

The paper introduces a new metric of synergy termed $\mathcal{S}_{\cap\cup}$, which is calculated as the mutual information between the entire set of predictors and the target variable, less the mutual information preserved when only redundant information is retained. This approach builds on the conceptualization of synergy as mutual information in excess of the "union" of its parts. Here, the union is defined by a minimization strategy that preserves correlations with single predictors while nullifying other dependencies.

Key Insights and Numerical Examples

Theoretical results and numerical simulations across various scenarios, including elementary logic gates like XOR and AND, underpin the theoretical soundness and practical applicability of $\mathcal{S}_{\cap\cup}$. Through these examples, the proposed measure shows:

• Invariance to duplicate predictors, ensuring a system's synergy measure doesn't artificially inflate due to repeated predictors.
• Capacity to differentiate between actual synergy and redundant synergies in cases where single predictors might otherwise dominate due to redundancy.

Implications and Future Directions

This advancement in quantifying synergistic mutual information stands to refine how information is understood in complex systems, transitioning from merely looking at redundancy and unique information to a focus on the composite contributions of predictors. This is particularly relevant in neural, genetic, and complex systems where interactions and dependencies among variables are critical.

Future developments could delve into the computational optimization of $\mathcal{S}_{\cap\cup}$ given its numerical heavy-lifting and explore its applicability in real-world datasets where complex variable interactions are prevalent. Moreover, establishing connections between information-theoretic synergy measures and physical systems could provide deeper insights into fundamental biological processes and machine learning algorithms.

Overall, this paper provides a comprehensive, mathematical, and practical contribution to the ongoing dialogue about synergy in complex systems, suggesting a robust foundation for future research and application in both theoretical and applied domains.