Posterior Consistency for Bayesian Inverse Problems through Stability and Regression Results

TL;DR

This paper proves posterior consistency for Bayesian inverse problems, demonstrating that as data becomes more precise, the posterior distribution concentrates around the true input, with applications to elliptic PDE inverse problems.

Contribution

It establishes posterior consistency for nonlinear inverse problems using stability and regression results, providing a rigorous foundation for Bayesian methods in such contexts.

Findings

Posterior measures concentrate around the true input as data accuracy improves.

Stability results for the inverse problem are key to establishing posterior consistency.

Posterior consistency depends on prior regularity, tail behavior, and small ball probabilities.

Abstract

In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise to the posterior distribution on the unknown input. In this setting we prove posterior consistency for nonlinear inverse problems: a sequence of data is considered, with diminishing fluctuations around a single truth and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around the truth used to generate the data. Posterior consistency justifies the use of the Bayesian approach very much in the same way as error bounds and convergence results for regularisation techniques do. As a guiding example, we consider the inverse problem of reconstructing the diffusion coefficient from noisy observations of the solution to an elliptic PDE in divergence form. This problem is approached by splitting the forward operator into the underlying continuum model and a simpler observation operator based on the output of the model. In general, these splittings allow us to conclude posterior consistency provided a deterministic stability result for the underlying inverse problem and a posterior consistency result for the Bayesian regression problem with the push-forward prior. Moreover, we prove posterior consistency for the Bayesian regression problem based on the regularity, the tail behaviour and the small ball probabilities of the prior.