  Estimating high-quality images while also quantifying their uncertainty are
two desired features in an image reconstruction algorithm for solving ill-posed
inverse problems. In this paper, we propose plug-and-play Monte Carlo (PMC) as
a principled framework for characterizing the space of possible solutions to a
general inverse problem. PMC is able to incorporate expressive score-based
generative priors for high-quality image reconstruction while also performing
uncertainty quantification via posterior sampling. In particular, we introduce
two PMC algorithms which can be viewed as the sampling analogues of the
traditional plug-and-play priors (PnP) and regularization by denoising (RED)
algorithms. We also establish a theoretical analysis for characterizing the
convergence of the PMC algorithms. Our analysis provides non-asymptotic
stationarity guarantees for both algorithms, even in the presence of
non-log-concave likelihoods and imperfect score networks. We demonstrate the
performance of the PMC algorithms on multiple representative inverse problems
with both linear and nonlinear forward models. Experimental results show that
PMC significantly improves reconstruction quality and enables high-fidelity
uncertainty quantification.
