Regularized Posteriors in Linear Ill‐Posed Inverse Problems

This is the authors' abstract. We don't add key points for this paper.

We study the Bayesian solution of a linear inverse problem in a separable Hilbert space setting with Gaussian prior and noise distribution. Our contribution is to propose a new Bayes estimator which is a linear and continuous estimator on the whole space and is stronger than the mean of the exact Gaussian posterior distribution which is only defined as a measurable linear transformation . Our estimator is the mean of a slightly modified posterior distribution called regularized posterior distribution . Frequentist consistency of our estimator and of the regularized posterior distribution is proved. A Monte Carlo study and an application to real data confirm good small‐sample properties of our procedure.