Noise, Denoise, Correct: MCMC Posterior Sampling with Diffusion Priors in Three Steps

2026-10-07

AI Summary

This work addresses the challenge of posterior sampling with diffusion priors under nonlinear, non-differentiable forward models by proposing a method termed Diffusion Waltz. The approach constructs Markov chain Monte Carlo (MCMC) proposal distributions through SDEdit-style noising and denoising, and incorporates observational information via a gradient-free ensemble Kalman update. A Metropolis-Hastings correction is subsequently applied to achieve exact posterior sampling without requiring prior density evaluation. Evaluated on Navier-Stokes initial condition recovery tasks, the proposed method significantly outperforms existing baselines across varying noise levels and degrees of nonlinearity, demonstrating both theoretical rigor and practical effectiveness.

Abstract

Pretrained diffusion models are powerful priors for inverse problems, but posterior sampling under nonlinear, non-differentiable forward models remain hard. We introduce diffusion waltz, an MCMC method using SDEdit-style noising-denoising as a proposal, corrected via Metropolis-Hastings for exact posterior sampling without prior evaluation. We further propose injecting observations into the proposal while preserving exactness, using a gradient-free ensemble Kalman update. On a non-differentiable Navier-Stokes initial condition recovery task, diffusion waltz outperforms existing baselines across different noise and nonlinearity regimes.

Problem

• inverse problems
• posterior sampling
• diffusion models
• non-differentiable forward models
• MCMC

Innovation

• Diffusion Priors
• MCMC Posterior Sampling
• Metropolis-Hastings
• Ensemble Kalman Update
• Inverse Problems