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 genera- tive 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), vari- ational auto-encoders (VAEs), normalizing flows and diffusion models (DMs), have demonstrated tremen- dous success in capturing detailed data representa- tions 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 uncertain- ties, especially in large-scale imaging applications. Building on these fundamental insights, posterior sampling techniques utilizing DMs have demon- strated remarkable efficiency and robustness, high- lighting their potential to effectively tackle complex.