On the Entropy-Based Localization of Inequality in Probability Distributions

Area of Science:

• Information Theory
• Statistical Analysis
• Data Science

Background:

• Scalar summaries often obscure localized disparities within probability distributions.
• Understanding the structure and concentration of inequality is crucial for analyzing heterogeneous systems.

Purpose of the Study:

• To introduce a novel method for localizing inequality within probability distributions.
• To demonstrate the method's applicability across various domains, including statistical and physical systems.

Main Methods:

• Applied a recursive Hahn decomposition to the degree of uniformity, derived from the exponential of Shannon entropy.
• Partitioned probability spaces into disjoint regions showing deviations from uniformity.
• Utilized canonical distributions (binomial, exponential) and hypothetical examples (disease contraction, loaded beam).

Main Results:

• Successfully localized inequality and identified structural disparities in both canonical and applied systems.
• Revealed targeted zones of epidemiological disparity in disease contraction data.
• Uncovered stress localization in a non-uniformly loaded beam, highlighting relevance to physical systems.

Conclusions:

• The recursive decomposition provides a multi-scale representation of informational non-uniformity.
• The framework offers structural insights into the emergence and localization of inequality.
• Potential implications for understanding entropy localization and dynamics of heterogeneous systems.