Bridging data-driven priors via the score function for posterior sampling -- Comparative review and experimental study

Researchers have proposed a unified framework for integrating diverse data-driven priors in Bayesian inverse problems through their score functions, applicable to image inpainting and super-resolution.

A paper submitted to arXiv on June 11, 2026, reviews how to unify data-driven priors in Bayesian inverse problems via score functions^[1]. The method integrates regularization-by-denoising, normalizing flow-based priors, score-based generative models, and convex-ridge regularizers into a sampling algorithm. The unified framework was applied to image inpainting, single image super-resolution, and restoring real images in a geological context, demonstrating its efficiency. Score function estimation, a key component, is tied to the quality of score-based generative models' output and can achieve minimax estimation rates by constraining the hypothesis space to a Sobolev ball^[2]. The proposed method uses score function estimators obtained by constraining the hypothesis class to a Sobolev ball, preventing overfitting. The framework can handle any posterior distribution defined by a broad class of score function-based priors.