Correcting misspecified score-based priors for inverse problems: An application to strong gravitational lensing

Score-based generative models have gained popularity as expressive, data-driven priors for complex, high-dimensional inverse problems. However, in many scientific applications, it is often difficult or even impossible to acquire samples from the true distribution to train these models, in which case a surrogate, e.g. a simulator, is often used to produce training samples, meaning that the learned prior could be misspecified. This, in turn, can bias the inferred posteriors, which limits the potential applicability of these models in real-world scenarios. In this work, we propose addressing this issue by iteratively training new priors with posterior samples from different sets of observations. We showcase the potential of this method on the problem of background image reconstruction in strong gravitational lensing. We show that posterior sampling becomes less biased after several updates, and the learned distribution is closer to the true prior.