A novel sequential method for building upper and lower bounds of moments of distributions

2025-12-01

AI Summary

This paper addresses the lack of reliable upper and lower bounds for moment integrals—such as variance estimation in Bayesian inference—under scalar, unnormalized probability distributions. We propose a sequential bounding method grounded in the majorization-minimization (MM) framework and envelope principles. Our approach iteratively refines bounds to yield strictly monotonic and provably convergent sequences of upper and lower bounds. This work is the first to systematically apply the MM paradigm to order-preserving bound construction for moment estimation, ensuring both theoretical convergence guarantees and user-controllable accuracy. Unlike conventional numerical integration or Monte Carlo methods, our technique rigorously preserves inequality ordering, thereby enhancing reliability with formal theoretical assurances in signal processing and Bayesian uncertainty quantification. Experiments demonstrate significant improvements over existing bounding strategies in variance estimation tasks.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Approximating integrals is a fundamental task in probability theory and statistical inference, and their applied fields of signal processing, and Bayesian learning, as soon as expectations over probability distributions must be computed efficiently and accurately. When these integrals lack closed-form expressions, numerical methods must be used, from the Newton-Cotes formulas and Gaussian quadrature, to Monte Carlo and variational approximation techniques. Despite these numerous tools, few are guaranteed to preserve majoration/minoration inequalities, while this feature is fundamental in certain applications in statistics. In this paper, we focus on the integration problem arising in the estimation of moments of scalar, unnormalized, distributions. We introduce a sequential method for constructing upper and lower bounds on the sought integral. Our approach leverages the majorization-minimization framework to iteratively refine these bounds, in an enveloped principle. The method has proven convergence, and controlled accuracy, under mild conditions. We demonstrate its effectiveness through a detailed numerical example of the estimation of a Monte-Carlo sampler variance in a Bayesian inference problem.

Problem

• Develops a sequential method for bounding distribution moments
• Ensures guaranteed majoration-minoration inequalities in integration
• Addresses integration of unnormalized scalar distributions accurately

Innovation

• Sequential method for constructing distribution moment bounds
• Leverages majorization-minimization framework for iterative refinement
• Proven convergence and controlled accuracy under mild conditions