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Regularized Posteriors in Linear Ill-Posed Inverse Problems

Authors


Anna Simoni, Department of Decision Sciences and IGIER, Università Bocconi, via G. Röntgen, 1 - 20136 Milano, Italy. E-mail: anna.simoni@unibocconi.it

Abstract

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.

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