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An Intrinsic Characterization of Shannon's and Rényi's Entropy
Martin Schlather1, Carmen Ditscheid1
1Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany.
Entropy (Basel, Switzerland)
|January 8, 2025
Summary
The Shannon entropy chain rule can be decomposed into additivity for cross-products and a new self-similarity property for single distributions. This finding extends to Rényi, min, and Hartley entropies, revealing intrinsic properties.
Area of Science:
- Information Theory
- Probability Theory
- Statistical Mechanics
Background:
- The Shannon entropy is a fundamental concept in information theory, quantifying uncertainty.
- Characterizations of entropy often rely on properties like the chain rule for hierarchical probability distributions.
Purpose of the Study:
- To decompose the Shannon entropy chain rule into fundamental components.
- To explore the implications of this decomposition for understanding entropy's intrinsic properties.
- To extend these findings to other related entropy measures.
Main Methods:
- Decomposition of the Shannon entropy chain rule into two distinct parts.
- Analysis of the properties of each component, particularly a proportionality relation for a single distribution.
- Application of analogous decomposition methods to Rényi entropy and its limits (min-entropy, Hartley entropy).
Main Results:
- The Shannon entropy chain rule is shown to consist of additivity for cross-products and a novel proportionality relation for single distributions.
- This proportionality relation suggests a self-similarity or intrinsic property of Shannon entropy.
- Similar decompositions and interpretations are established for Rényi, min-, and Hartley entropies.
Conclusions:
- The decomposition provides a new perspective on the structure of Shannon entropy and its related measures.
- The identified self-similarity offers deeper insights into the fundamental nature of information uncertainty.
- This work paves the way for further research into the structural properties of various entropy measures.
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