Proposal Description

Inverse problems arise naturally in scientific applications in physics, imaging, engineering, and many other areas. They are concerned with inference on a possibly infinite dimensional parameter from indirect measurements, often observed with independent measurement noise. A prototypical example is the parameter identification in a partial differential equation (PDE). The Bayesian approach provides a systematic and widely used methodology for solving inverse problems with automatic uncertainty quantification and has become more feasible with the recent increase in computational resources.
 
A practitioner needs to specify a numerically tractable likelihood-function and a reasonable prior distribution, and in return obtains by Bayes’ theorem a posterior distribution that can be used, in principle, for all inferential tasks. Compared to classical inverse problem techniques, there is no explicit inversion of operators. While the simplicity of this approach is attractive and the de-facto standard in many applications such as weather forecasting and astrostatistics, without statistical guarantees the posterior may not be informative about the statistical target, even for infinite data. Moreover, since the posterior is rarely known in closed form we need to rely on sampling algorithms. Without algorithmic guarantees, however, the results cannot be trusted. For low-dimensional linear inverse problems, the Bayesian approach is well understood, but its study for more complex, high-dimensional or non-linear inverse problems is a very active research field with a wealth of open theoretical and practical challenges.
 
The goal of the session is to provide a platform for young and mid-career researchers to exchange ideas about some of the most pressing challenges in Bayesian inverse problems. The speakers are selected for their excellence and to have a variety of backgrounds and interests related to three main topics. First, techniques for nonlinear inverse problems with PDEs and diffusion processes for dependent data. Second, computational guarantees for sampling-based inference procedures. And third, Bayesian methodology and the effects when choosing different prior distributions.
 
While Bayesian inference is an established field in statistical theory, Bayesian inverse problems form a more recent, but by now indispensable addition to modern Data Science. I believe it is crucial to highlight this importance with a session at the most important statistics conference.