Generative modelling meets Bayesian inference: a new paradigm for inverse problems.

Area of Science:

• Computational Mathematics
• Machine Learning
• Statistical Inference

Background:

• Traditional Bayesian priors struggle with complex data distributions.
• Deep generative models (DGMs) excel at capturing intricate data representations.
• DGMs offer superior accuracy and perceptual realism compared to conventional methods.

Purpose of the Study:

• To explore Bayesian inverse problems using data-driven priors from DGMs.
• To investigate the synergy between generative modeling and Bayesian inference.
• To highlight advancements in uncertainty quantification for inverse problems.

Main Methods:

• Utilizing deep generative models (DGMs) including GANs, VAEs, normalizing flows, and diffusion models (DMs).
• Formulating Bayesian inverse problems with conditional Wasserstein GANs.
• Applying posterior sampling techniques with DMs for efficiency and robustness.

Main Results:

• DGMs provide accurate priors that capture complex data geometry.
• Conditional Wasserstein GANs improve uncertainty quantification in large-scale imaging.
• Diffusion models demonstrate efficient and robust posterior sampling for inverse problems.

Conclusions:

• Deep generative priors overcome limitations of traditional Bayesian methods.
• This convergence enriches the theoretical and practical aspects of Bayesian inversion.
• The paradigm shift offers profound implications for scientific and engineering applications.