Bayesian inverse problems with Gaussian priors

Abstract

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