Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators

2026-02-24

AI Summary

This work proposes Proximal-IMH to address the computational expense of exact posterior sampling in Bayesian inverse problems. Within an independent Metropolis–Hastings framework, the method introduces a proximal correction mechanism that locally refines proposal samples by solving an auxiliary optimization problem near approximate posterior draws, thereby balancing model fidelity and reference point stability. Theoretical analysis demonstrates that this correction effectively reduces the discrepancy between the approximate and true posterior distributions, leading to improved acceptance rates and enhanced Markov chain mixing. Experimental results on nonlinear inverse problems—featuring multimodal structures and data-driven priors—show that Proximal-IMH significantly outperforms existing variants of independent Metropolis–Hastings samplers.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• 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

Abstract

We consider the problem of sampling from a posterior distribution arising in Bayesian inverse problems in science, engineering, and imaging. Our method belongs to the family of independence Metropolis-Hastings (IMH) sampling algorithms, which are common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but may have significant bias, we introduce Proximal-IMH, a scheme that removes this bias by correcting samples from the approximate posterior through an auxiliary optimization problem. This yields a local adjustment that trades off adherence to the exact model against stability around the approximate reference point. For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, thereby improving acceptance rates and mixing. The method applies to both linear and nonlinear input-output operators and is particularly suitable for inverse problems where exact posterior sampling is too expensive. We present numerical experiments including multimodal and data-driven priors with nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.

Problem

• Bayesian inverse problems
• posterior sampling
• approximate posterior
• Metropolis-Hastings
• bias correction

Innovation

• Proximal-IMH
• Bayesian inverse problems
• Independence Metropolis-Hastings
• posterior correction
• approximate sampling