Path integral methods for Bayesian inference

Inverse problems in infinite dimensions are ubiquitously encountered across the scientific disciplines. These problems are defined by the need to reconstruct continuous fields from incomplete, noisy measurements, which oftentimes leads to ill-posed problems. Almost universally, the solutions to these problems are constructed under, or can be viewed as a limit case of, a Bayesian framework. However, in the infinite-dimensional setting, the theory is largely restricted to the Gaussian case due to some technical difficulties. The most notable being that the Lebesgue measure does not exist on infinite-dimensional spaces. As a result, we often resort to Gaussian measures so that the prior and resulting posterior remain well-defined. As an alternative, we explore the use of the Feynman path integral formalism for Bayesian inference. By posing inverse problems with path integrals, the resulting forms resemble the finite-dimensional setting, which allows for intuitive techniques to be derived. In this talk, we discuss the theory, numerical methods, and some real-world applications involved under this viewpoint.

Bio
https://www.linkedin.com/in/alex-alberts-930a4b262/