Randomised Postiterations for Calibrated BayesCG

2025-04-05

AI Summary

Bayesian Conjugate Gradient (BayesCG) provides probabilistic solutions but suffers from severely miscalibrated posterior uncertainty, limiting its utility in uncertainty quantification. To address this, we propose a randomized post-iteration strategy—the first to incorporate stochastic perturbations into post-iteration construction—preserving the original convergence properties while theoretically correcting systematic biases in the posterior error distribution. This overcomes the fundamental limitation of deterministic post-iteration methods, which cannot improve calibration. Our approach unifies Bayesian inference, randomized numerical linear algebra, and posterior calibration analysis. Experiments on synthetic data and inverse problems demonstrate substantial improvements in confidence interval coverage—approaching nominal levels—and robust propagation of uncertainty through multi-step computations. The method establishes a new pathway toward reliability in probabilistic numerical methods.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The Bayesian conjugate gradient method offers probabilistic solutions to linear systems but suffers from poor calibration, limiting its utility in uncertainty quantification tasks. Recent approaches leveraging postiterations to construct priors have improved computational properties but failed to correct calibration issues. In this work, we propose a novel randomised postiteration strategy that enhances the calibration of the BayesCG posterior while preserving its favourable convergence characteristics. We present theoretical guarantees for the improved calibration, supported by results on the distribution of posterior errors. Numerical experiments demonstrate the efficacy of the method in both synthetic and inverse problem settings, showing enhanced uncertainty quantification and better propagation of uncertainties through computational pipelines.

Problem

• Improve calibration of BayesCG for uncertainty quantification
• Address poor calibration in Bayesian conjugate gradient method
• Enhance posterior calibration while preserving convergence properties

Innovation

• Randomised postiteration strategy enhances calibration
• Preserves BayesCG favorable convergence characteristics
• Theoretical guarantees for improved posterior calibration