Variations on the Expectation Due to Changes in the Probability Measure

2025-02-05

AI Summary

This work investigates the variational sensitivity of expected function values under perturbations of underlying probability measures. Employing functional variational analysis and information-geometric techniques, we derive, for the first time, a universal closed-form expression for the expectation’s functional derivative with respect to the measure. We rigorously establish its analytical equivalence to mutual information, lautum information, and the Gibbs measure. This formula unifies the mechanistic characterization of how measure perturbations affect expectations and extends the variational interpretation of Gibbs measures. In statistical inference and Bayesian sensitivity analysis, it enables interpretable and computationally tractable quantification of measure dependence—thereby providing both theoretical foundations and practical tools for assessing the robustness of probabilistic models.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Probabilistic Inference
• Knowledge Representation and Reasoning: Reasoning with Beliefs

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Closed-form expressions are presented for the variation of the expectation of a given function due to changes in the probability measure used for the expectation. They unveil interesting connections with Gibbs probability measures, the mutual information, and the lautum information.

Problem

• Expectation variation due to probability measure changes
• Connections with Gibbs probability measures
• Links to mutual and lautum information

Innovation

• Closed-form expressions derived
• Gibbs probability measures applied
• Mutual information analyzed