Reconsideration of Information-Theoretic Principles-Perspective from the Dual Probability Distribution.

Area of Science:

• Information Theory
• Probability Theory
• Statistical Mechanics

Background:

• Information-theoretic principles, such as maximum entropy, are fundamental in statistical inference.
• The Massieu potential is a key concept related to these principles.
• Understanding these principles is crucial for various fields, including machine learning and physics.

Purpose of the Study:

• To reconsider information-theoretic principles from the novel perspective of dual probability distributions.
• To establish a connection between the Massieu potential and Kullback-Leibler divergence using dual distributions.
• To explore the implications of this new perspective for data analysis.

Main Methods:

• Introduction of the dual probability distribution via Sanov's Lemma for multinomial distributions.
• Rewriting the Massieu potential as Kullback-Leibler divergence between dual distributions.
• Expressing the dual potential as a cumulant generating function.

Main Results:

• The dual correspondence between probability distributions becomes asymptotically manifest.
• The Massieu potential is reformulated as the Kullback-Leibler divergence involving the dual distribution.
• The dual potential is shown to be the cumulant generating function with respect to the dual reference distribution.

Conclusions:

• The dual probability distribution offers a new and insightful perspective on information-theoretic principles.
• This framework naturally connects to data sampling, suggesting significant potential for data analysis.
• The findings pave the way for new methodologies in statistical inference and machine learning.