On Stability and Robustness of Diffusion Posterior Sampling for Bayesian Inverse Problems

2026-02-02

AI Summary

Diffusion models in Bayesian inverse problems often rely on assumed likelihoods, yet the relationship between these assumptions and reconstruction quality remains unclear, and their performance is fragile under likelihood misspecification. This work presents the first systematic analysis of the stability of diffusion-based posterior samplers, revealing their sensitivity to likelihood misspecification. Building on this insight, we propose a theoretically grounded robust posterior sampling method that integrates seamlessly with existing gradient-guided sampling frameworks. Extensive experiments on scientific inverse problems and natural image reconstruction tasks demonstrate the efficacy of our approach, which consistently achieves significantly improved reconstruction stability and performance—even under severe likelihood misspecification.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Diffusion models have recently emerged as powerful learned priors for Bayesian inverse problems (BIPs). Diffusion-based solvers rely on a presumed likelihood for the observations in BIPs to guide the generation process. However, the link between likelihood and recovery quality for BIPs is unclear in previous works. We bridge this gap by characterizing the posterior approximation error and proving the \emph{stability} of the diffusion-based solvers. Meanwhile, an immediate result of our findings on stability demonstrates the lack of robustness in diffusion-based solvers, which remains unexplored. This can degrade performance when the presumed likelihood mismatches the unknown true data generation processes. To address this issue, we propose a simple yet effective solution, \emph{robust diffusion posterior sampling}, which is provably \emph{robust} and compatible with existing gradient-based posterior samplers. Empirical results on scientific inverse problems and natural image tasks validate the effectiveness and robustness of our method, showing consistent performance improvements under challenging likelihood misspecifications.

Problem

• Bayesian inverse problems
• diffusion models
• posterior sampling
• robustness
• likelihood misspecification

Innovation

• stability
• robustness
• diffusion models
• Bayesian inverse problems
• posterior sampling