Posterior sampling has been shown to be a powerful Bayesian approach for solving imaging inverse problems. The recent plug-and-play unadjusted Langevin algorithm (PnP-ULA) has emerged as a promising method for Monte Carlo sampling and minimum mean squared error (MMSE) estimation by combining physical measurement models with deep-learning priors specified using image denoisers. However, the intricate relationship between the sampling distribution of PnP-ULA and the mismatched data-fidelity and denoiser has not been theoretically analyzed. We address this gap by proposing a posterior-L2 pseudometric and using it to quantify an explicit error bound for PnP-ULA under mismatched posterior distribution. We numerically validate our theory on several inverse problems such as sampling from Gaussian mixture models and image deblurring. Our results suggest that the sensitivity of the sampling distribution of PnP-ULA to a mismatch in the measurement model and the denoiser can be precisely characterized.

翻译：后验采样已被证明是解决成像逆问题的强大贝叶斯方法。近年提出的即插即用非调整朗之万算法（PnP-ULA），通过将物理测量模型与基于图像去噪器的深度学习先验相结合，已成为蒙特卡洛采样和最小均方误差（MMSE）估计领域的一种前景广阔的方法。然而，PnP-ULA的采样分布与失配的数据保真项及去噪器之间的复杂关系尚未得到理论分析。为填补这一空白，我们提出了一种后验L2伪度量，并据此量化了在失配后验分布条件下PnP-ULA的显式误差界。我们通过高斯混合模型采样和图像去模糊等多个逆问题对理论进行数值验证。结果表明，PnP-ULA的采样分布对测量模型与去噪器失配的敏感性可以得到精确刻画。