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

This special issue addresses Bayesian inverse problems using data-driven priors derived from deep generative models (DGMs) and the convergence of generative modelling techniques and Bayesian inference methods. Conventional Bayesian priors often fail to accurately capture the properties and the underlying geometry of complex, real-world data distributions. In contrast, deep generative models (DGMs), which include generative adversarial networks (GANs), variational auto-encoders (VAEs), normalizing flows and diffusion models (DMs), have demonstrated tremendous success in capturing detailed data representations learned directly from empirical observations. As a result, these models produce priors endowed with superior accuracy, increased perceptual realism and enhanced capacities for uncertainty quantification within inverse problem contexts. This paradigm emerged in the late 2010s, when pioneering efforts were made to explicitly formulate Bayesian inverse problems using conditional Wasserstein generative adversarial networks (GANs). These advances have greatly improved methods for quantifying uncertainties, especially in large-scale imaging applications. Building on these fundamental insights, posterior sampling techniques utilizing DMs have demonstrated remarkable efficiency and robustness, highlighting their potential to effectively tackle complex and diverse inverse problems. The articles collected herein provide essential theoretical breakthroughs and significant algorithmic innovations, collectively demonstrating how deep generative priors mitigate traditional limitations and profoundly enrich both the practical applicability and theoretical foundations of Bayesian inversion.This article is part of the theme issue 'Generative modelling meets Bayesian inference: a new paradigm for inverse problems'.