  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.
