Bridging diffusion posterior sampling and Monte Carlo methods: a survey

Area of Science:

• Generative modeling
• Bayesian inference
• Scientific computing

Background:

• Diffusion models excel at generating accurate samples from complex distributions, establishing them as foundational in generative modeling.
• Bayesian inverse problems are critical in various scientific fields, often requiring robust prior information for effective solutions.
• Pre-trained diffusion models offer a powerful new avenue for addressing these inverse problems by acting as sophisticated priors.

Purpose of the Study:

• To provide a comprehensive review of current methodologies for solving Bayesian inverse problems using pre-trained diffusion models and Monte Carlo techniques.
• To elucidate the core mechanisms, particularly the 'twisting' of intermediate distributions, that enable these models to guide simulations toward posterior distributions.
• To detail the integration of various Monte Carlo methods for efficient sampling from these adapted diffusion processes.

Main Methods:

• Leveraging pre-trained diffusion models as priors without requiring task-specific fine-tuning.
• Employing a 'twisting' mechanism to modify intermediate distributions within the diffusion process.
• Integrating diverse Monte Carlo sampling strategies to draw samples from the guided posterior distributions.

Main Results:

• Demonstrated effectiveness of pre-trained diffusion models in solving Bayesian inverse problems.
• Showcased the 'twisting' mechanism as a key component for adapting diffusion models to posterior inference.
• Illustrated the synergistic application of diffusion models with various Monte Carlo methods for robust sampling.

Conclusions:

• The combination of pre-trained diffusion models and Monte Carlo methods presents a powerful, training-free approach for Bayesian inverse problems.
• The 'twisting' technique is central to adapting generative priors for posterior inference in inverse problems.
• This paradigm shift, merging generative modeling with Bayesian inference, opens new possibilities for scientific discovery.