Bayesian inverse problems with partial observations

TL;DR

This paper analyzes a Bayesian approach to linear inverse problems with partial observations, showing how posterior distributions contract around the true parameter at optimal rates and how prior choices affect credible set coverage.

Contribution

It introduces a nonparametric Bayesian framework using Fourier transforms for inverse problems and characterizes posterior contraction and coverage properties.

Findings

Posterior contracts at optimal rates depending on smoothness and ill-posedness.

Coverage of Bayesian credible sets depends on prior smoothness, with oversmoothing leading to zero coverage.

Numerical examples illustrate theoretical results.

Abstract

We study a nonparametric Bayesian approach to linear inverse problems under discrete observations. We use the discrete Fourier transform to convert our model into a truncated Gaussian sequence model, that is closely related to the classical Gaussian sequence model. Upon placing the truncated series prior on the unknown parameter, we show that as the number of observations $n\rightarrow\infty,$ the corresponding posterior distribution contracts around the true parameter at a rate depending on the smoothness of the true parameter and the prior, and the ill-posedness degree of the problem. Correct combinations of these values lead to optimal posterior contraction rates (up to logarithmic factors). Similarly, the frequentist coverage of Bayesian credible sets is shown to be dependent on a combination of smoothness of the true parameter and the prior, and the ill-posedness of the problem. Oversmoothing priors lead to zero coverage, while undersmoothing priors produce highly conservative results. Finally, we illustrate our theoretical results by numerical examples.