Variations on the Expectation Due to Changes in the Probability Measure.

Area of Science:

• Information Theory
• Probability Theory
• Statistical Mechanics

Background:

• Understanding how changes in probability distributions affect statistical properties is crucial in various scientific fields.
• Existing methods may lack closed-form solutions for analyzing these variations.

Purpose of the Study:

• To derive closed-form expressions for the variation of function expectations under probability measure changes (probability distribution drifts).
• To explore the theoretical implications and connections of these expressions within information theory and statistical mechanics.

Main Methods:

• Derivation of closed-form analytical expressions.
• Mathematical analysis of probability measure variations.
• Connections to established information-theoretic quantities.

Main Results:

• Novel closed-form expressions quantifying expectation variation due to probability distribution drifts.
• Demonstrated relationships between these expressions and Gibbs probability measures.
• Identified connections to information projections and Pythagorean identities for relative entropy, mutual information, and lautum information.

Conclusions:

• The derived expressions provide a powerful tool for analyzing probability distribution drifts.
• The study highlights fundamental connections between expectation variations and core concepts in information theory.
• These findings can advance research in areas sensitive to distributional changes, such as machine learning and statistical physics.