A pedestrian's approach to Bayesian methods for inverse problems

Abstract

In the past two decades, the Bayesian approach to inverse problems in infinite dimensional spaces has seen significant developments in theory, Markov chain Monte Carlo (MCMC) methods, and in applications. The Bayesian approach in finite dimension produces posterior distributions of finite dimensional random variables. In infinite dimensions, the desired objects are random fields and their laws as measures on Hilbert or Banach spaces. There are many textbooks and tutorials available that explain Bayesian methods and MCMC in finite dimensions. To enter the field of Bayesian inverse problems in infinite dimensions, a newcomer has to engage with a quite diverse and extensive literature. The purpose of this article is to provide a basic path to the subject that is easily accessible to a reader with general knowledge in probability and mathematical analysis. We start with a swift introduction to the finite dimensional case. The infinite dimensional case is introduced for Gaussian priors on Hilbert spaces and Gaussian-type likelihood functions. In this situation, mere continuity of the forward map suffices to prove that the posterior measure is well-defined. We also give some new and simple stability results. Two application examples are elaborated, one finite dimensional, one infinite dimensional.