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On the Entropy-Based Localization of Inequality in Probability Distributions.

Rajeev Rajaram1, Nathan Ritchey1, Brian Castellani2

  • 1Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA.

Entropy (Basel, Switzerland)
|September 27, 2025
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We developed a new method to pinpoint inequality in probability distributions using recursive decomposition. This technique reveals where and how inequality concentrates, offering structural insights into data heterogeneity.

Keywords:
Hahn decompositionShannon entropydegree of uniformity

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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.