Bridging diffusion posterior sampling and Monte Carlo methods: a survey

Diffusion models enable the synthesis of highly accurate samples from complex distributions and have become foundational in generative modelling. Recently, they have demonstrated significant potential for solving Bayesian inverse problems by serving as priors. This review offers a comprehensive overview of current methods that leverage pre-trained diffusion models alongside Monte Carlo methods to address Bayesian inverse problems without requiring additional training. We show that these methods primarily employ a twisting mechanism for the intermediate distributions within the diffusion process, guiding the simulations towards the posterior distribution. We describe how various Monte Carlo methods are then used to aid in sampling from these twisted distributions.This article is part of the theme issue 'Generative modelling meets Bayesian inference: a new paradigm for inverse problems'.