• functional data;
  • Gaussian process priors;
  • inverse problems;
  • measurable linear transformation;
  • posterior consistency;
  • Tikhonov regularization

Abstract.  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.