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Briding Diffusion Posterior Sampling and Monte Carlo methods: a survey

Yazid Janati, Alain Durmus, Jimmy Olsson, Eric Moulines

TL;DR

This survey addresses the challenge of sampling from posterior distributions in Bayesian inverse problems when using pre-trained diffusion priors. It catalogs gradient-guided and Monte Carlo-based strategies that twist diffusion intermediates toward the posterior, including DPS, PiGDM, and various sequence-based samplers (SMC, MCMC, VI). It details concrete inpainting-specific methods (Replacement, SMCDiff, MCGDiff) and theoretical insights such as contraction bounds and optimal proposals, while discussing practical trade-offs in computation and memory. The work highlights open questions on distribution sequencing, sampling scheme selection, and the quest for scalable, high-performance DPS-type potentials with reduced cost and memory demands.

Abstract

Diffusion models enable the synthesis of highly accurate samples from complex distributions and have become foundational in generative modeling. 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 \emph{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 \emph{twisting} mechanism for the intermediate distributions within the diffusion process, guiding the simulations toward the posterior distribution. We describe how various Monte Carlo methods are then used to aid in sampling from these twisted distributions.

Briding Diffusion Posterior Sampling and Monte Carlo methods: a survey

TL;DR

This survey addresses the challenge of sampling from posterior distributions in Bayesian inverse problems when using pre-trained diffusion priors. It catalogs gradient-guided and Monte Carlo-based strategies that twist diffusion intermediates toward the posterior, including DPS, PiGDM, and various sequence-based samplers (SMC, MCMC, VI). It details concrete inpainting-specific methods (Replacement, SMCDiff, MCGDiff) and theoretical insights such as contraction bounds and optimal proposals, while discussing practical trade-offs in computation and memory. The work highlights open questions on distribution sequencing, sampling scheme selection, and the quest for scalable, high-performance DPS-type potentials with reduced cost and memory demands.

Abstract

Diffusion models enable the synthesis of highly accurate samples from complex distributions and have become foundational in generative modeling. 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 \emph{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 \emph{twisting} mechanism for the intermediate distributions within the diffusion process, guiding the simulations toward the posterior distribution. We describe how various Monte Carlo methods are then used to aid in sampling from these twisted distributions.
Paper Structure (20 sections, 90 equations, 1 figure)

This paper contains 20 sections, 90 equations, 1 figure.

Table of Contents

  1. Posterior sampling for denoising diffusion priors
  2. Bayesian inverse problems
  3. Denoising Diffusion models
  4. Posterior sampling
  5. Gradient guidance
  6. A sequence-of-distributions perspective
  7. Sequential Monte Carlo methods
  8. Markov chain Monte Carlo and variational inference methods
  9. Split Gibbs sampling
  10. Mixture Guided Posterior Sampling
  11. Divide-and-Conquer Posterior Sampling (DCPS)
  12. Midpoint Guidance Posterior Sampling
  13. The inpainting problem
  14. The Replacement method
  15. SMCDiff: Sequential Monte Carlo Diffusion algorithm
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Progressive reconstruction of degraded images using a diffusion model. The figure demonstrates the denoising process over time ($k$ steps) for three different tasks: inpainting (top row), motion deblurring (middle row), and super-resolution (bottom row). As the diffusion process unfolds, the model progressively refines the image, recovering details from an initial noisy state to a high-quality reconstruction.