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The Information Loss of a Stochastic Map
James Fullwood1, Arthur J Parzygnat2
1School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China.
Entropy (Basel, Switzerland)
|August 27, 2021
Summary
This study extends information theory by introducing a stochastic characterization of Shannon information loss. It defines conditional entropy and conditional information loss, with applications in Markov categories and entropic Bayes
Area of Science:
- Information Theory
- Category Theory
- Probability Theory
Background:
- The Baez-Fritz-Leinster (BFL) characterization provides a framework for understanding Shannon information loss.
- Measure-preserving functions are fundamental in probability and information theory.
- Existing information measures lack functorial properties in certain mathematical structures.
Purpose of the Study:
- To develop a stochastic extension of the BFL characterization for information loss.
- To introduce and define 'conditional information loss' as a novel information-theoretic measure.
- To explore the semi-functorial properties of these information measures within Markov categories.
Main Methods:
- Stochastic extension of the Baez-Fritz-Leinster characterization.
- Introduction of the concept of semi-functoriality in Markov categories.
- Development of an entropic Bayes' rule for information measures.
Main Results:
- The stochastic extension recovers conditional entropy and introduces conditional information loss.
- Conditional information loss and conditional entropy are shown to be semi-functorial.
- A characterization of conditional entropy is provided using the entropic Bayes' rule.
Conclusions:
- The proposed framework offers a generalized understanding of information loss and conditional entropy.
- Semi-functoriality provides a useful structure for information measures in probabilistic settings.
- The entropic Bayes' rule offers a new perspective on conditional entropy.
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