Bayesian inverse problems with Gaussian priors

TL;DR

This paper analyzes how Gaussian priors influence the contraction rates and coverage properties of posterior distributions in nonparametric inverse problems, highlighting the importance of prior smoothness and scale.

Contribution

It provides theoretical results on posterior contraction rates and coverage in nonparametric inverse problems with Gaussian priors, including conditions for minimax optimality.

Findings

Posterior contracts at a rate depending on prior and true parameter smoothness.

Coverage of credible sets varies with prior smoothness, from zero to conservative.

Numerical illustration with function recovery from noisy primitive observations.

Abstract

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive.