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Conditional Rényi Divergences and Horse Betting
Cédric Bleuler1, Amos Lapidoth1, Christoph Pfister1
1Signal and Information Processing Laboratory, ETH Zurich, 8092 Zurich, Switzerland.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
A new conditional Rényi divergence is introduced, enhancing the horse betting problem analysis. This new divergence is key for side information, leading to a universal strategy maximizing gambler utility without prior knowledge.
Area of Science:
- Information Theory
- Probability Theory
- Game Theory
Background:
- Conditional Rényi divergences are crucial for quantifying dependence.
- Existing measures like Csiszár's and Sibson's have limitations.
- Horse betting problems offer a practical motivation for developing new divergence measures.
Purpose of the Study:
- Introduce and analyze a novel conditional Rényi divergence.
- Compare its properties with existing conditional divergences under data processing.
- Develop a universal betting strategy maximizing utility with side information.
Main Methods:
- Mathematical analysis of conditional Rényi divergences.
- Comparison of divergence properties under data processing.
- Application of divergence measures to a generalized horse betting problem with power-mean utility functions.
Main Results:
- A new conditional Rényi divergence is defined and its properties studied.
- The new divergence leads to the Lapidoth-Pfister mutual information and relates to Arimoto-Rényi conditional entropy.
- A universal strategy is presented that maximizes utility for independent and identically distributed races.
Conclusions:
- The new conditional Rényi divergence offers advancements in dependence measure theory.
- It provides operational meaning to Lapidoth-Pfister mutual information in settings with side information.
- The developed universal strategy offers practical implications for betting and decision-making under uncertainty.
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