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Updated: Oct 25, 2025
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Published on: September 3, 2021
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ϕ-Informational Measures: Some Results and Interrelations
Steeve Zozor1, Jean-François Bercher2
1GIPSA-Lab, CNRS, Grenoble INP, University Grenoble Alpes, 38000 Grenoble, France.
Entropy (Basel, Switzerland)
|August 6, 2021
Summary
This study introduces extended informational measures, including state-dependent ϕ-entropies, by generalizing Fisher information and moments using escort distributions. These new measures unify and extend classical results in information theory.
Area of Science:
- Information Theory
- Mathematical Physics
- Statistical Inference
Background:
- Classical information theory relies on measures like entropy and Fisher information.
- Generalizations are needed to capture more complex statistical structures.
Purpose of the Study:
- To introduce and analyze extended informational measures based on convex functions.
- To generalize Fisher information and moments using escort distributions.
- To explore the role of maximum entropy distributions in this extended framework.
Main Methods:
- Definition of escort distributions linked to convex functions (ϕ).
- Introduction of state-dependent ϕ-entropies via the inverse maximum entropy problem.
- Examination of interrelations between generalized informational measures.
Main Results:
- Development of generalized Fisher information and moments.
- Establishment of state-dependent ϕ-entropies.
- Generalization of the Cramér-Rao inequality and de Bruijn identity.
- Demonstration that maximum entropy distributions are central.
Conclusions:
- The proposed extended informational measures provide a unified framework.
- Classical results are recovered as special cases of the generalized framework.
- This work broadens the scope of information-theoretic analysis.
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