Analytical Formulation of Continuous Redundancy Measure Based on Shared Exclusions

Core Concepts

Analytical formulation for continuous redundancy measure based on shared exclusions.

Abstract

The content introduces a novel analytic formulation for continuous redundancy, bridging the gap for a continuous PID measure. It discusses the importance of statistical dependencies and the theory of Partial Information Decomposition (PID). The paper presents a nearest-neighbor based estimator for continuous PID and showcases its effectiveness in a simulated energy management system. The analytical formulation is detailed, providing insights into the intricate dependencies between variables.

Introduction to Statistical Dependencies
Theory of Partial Information Decomposition (PID)
Novel Analytic Formulation for Continuous Redundancy
Nearest-Neighbor Based Estimator for Continuous PID
Application to Energy Management System
Bridging the Gap for Continuous PID Measures

Stats

"The main contributions of this paper are (1) the introduction of a tractable analytical PID definition inspired by the measure-theoretic definition of continuous Isx ∩ from Schick-Poland et al. [34] and its application to simple theoretical examples in Section II B, (2) the development of an estimator for the associated redundancy measure, which draws on concepts of the k-nearest-neighbours based estimator for mutual information by Kraskov et al. [35] in Section II C and (3) the demonstration of the efficacy of our continuous Isx ∩ measure in uncovering variable dependencies in data from an energy management system in Section IV."

Quotes

"Describing statistical dependencies is foundational to empirical scientific research."
"This work bridges the gap between the measure-theoretically postulated existence proofs for a continuous Isx ∩ and its practical application to real-world scientific problems."

Key Insights Distilled From

Partial Information Decomposition for Continuous Variables based on Shared Exclusions

by David A. Ehr... at arxiv.org 03-28-2024

https://arxiv.org/pdf/2311.06373.pdf

Deeper Inquiries

How can the concept of shared exclusions in probability space be practically applied in other scientific domains

The concept of shared exclusions in probability space, as applied in the context of Partial Information Decomposition (PID), can have practical applications in various scientific domains. For example, in neuroscience, shared exclusions can help in understanding how different neural variables contribute to the overall information processing in the brain. By identifying the unique, redundant, and synergistic information shared between neural variables, researchers can gain insights into the underlying mechanisms of brain function and information flow. Similarly, in machine learning, shared exclusions can aid in feature selection and dimensionality reduction by highlighting the specific contributions of different features to the overall predictive power of a model. By quantifying the shared exclusions between input variables and the target variable, machine learning algorithms can be optimized for better performance and interpretability.

What are the potential limitations of using a nearest-neighbor based estimator for continuous PID measures

While the nearest-neighbor based estimator for continuous PID measures, such as the one proposed in the study, offers advantages in terms of variance-bias characteristics, there are potential limitations to consider. One limitation is the sensitivity of the estimator to the choice of the parameter k, which determines the number of nearest neighbors considered in the estimation process. A suboptimal choice of k can lead to biased estimates of the information-theoretic quantities, especially in cases where the underlying probability distribution is complex or high-dimensional. Additionally, the nearest-neighbor based estimator may struggle with capturing non-linear dependencies or intricate relationships between variables, as it relies on local density estimation that may not fully capture the global structure of the data. Furthermore, the computational complexity of the estimator can increase significantly with larger datasets, making it less efficient for analyzing big data sets or real-time applications.

How can the findings of this study impact the development of analytical tools for information theory in complex systems

The findings of this study have the potential to significantly impact the development of analytical tools for information theory in complex systems. By introducing a novel analytical formulation for continuous redundancy based on shared exclusions in probability space, the study bridges the gap between theoretical postulates and practical applications of PID measures for continuous variables. This development opens up new possibilities for analyzing and quantifying dependencies in complex systems across various scientific disciplines. The proposed estimator for continuous PID measures provides a practical and interpretable approach to uncovering the unique, redundant, and synergistic information shared between variables in continuous systems. This can lead to advancements in understanding the underlying structures and interactions in complex systems, paving the way for more sophisticated data analysis techniques and information processing methodologies.