Regularized Posteriors in Linear Ill-Posed Inverse Problems

• Activity Number: 308
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2008 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Anna Simoni*+ and Jean-Pierre Florens
• Companies: Toulouse School of Economics (GREMAQ) and Toulouse School of Economics (GREMAQ - IDEI)
• Keywords: Gaussian Process ; Conditional Probability ; Tikhonov Regularization ; Sufficient Statistics ; Posterior Consistency

Abstract

This paper studies Bayesian solution for a signal-noise problem in which we observe a noisy transformation of the infinite dimensional parameter that we want to recover. Actually, we deal with a functional equation in Polish spaces which is ill-posed because of compactness of the operator characterizing it. We follow a Bayesian approach, hence the solution is the posterior distribution of the parameter of interest. To get rid of continuity problems caused by infinite dimension of the spaces, we compute a regularized version of the posterior distribution and we guess it is solution of the inverse problem. Frequentist consistency is proved under regularity conditions of the true parameter. Finally, we extend our analysis to the case with unknown operator and the case with different operators for each observation. Monte Carlo simulations confirm good properties of our estimator.