Regularizing priors for linear inverse problems

Jean-Pierre Florens and Anna Simoni

No 621, IDEI Working Papers from Institut d'Économie Industrielle (IDEI), Toulouse

Abstract: We consider statistical linear inverse problems in Hilbert spaces of the type ˆ Y = Kx + U where we want to estimate the function x from indirect noisy functional observations ˆY . In several applications the operator K has an inverse that is not continuous on the whole space of reference; this phenomenon is known as ill-posedness of the inverse problem. We use a Bayesian approach and a conjugate-Gaussian model. For a very general specification of the probability model the posterior distribution of x is known to be inconsistent in a frequentist sense. Our first contribution consists in constructing a class of Gaussian prior distributions on x that are shrinking with the measurement error U and we show that, under mild conditions, the corresponding posterior distribution is consistent in a frequentist sense and converges at the optimal rate of contraction. Then, a class ^ of posterior mean estimators for x is given. We propose an empirical Bayes procedure for selecting an estimator in this class that mimics the posterior mean that has the smallest risk on the true x.

Date: 2010
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Related works:
Journal Article: REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS (2016)
Working Paper: REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS (2016)
Working Paper: Regularizing Priors for Linear Inverse Problems (2013)
Working Paper: Regularizing Priors for Linear Inverse Problems (2013)
Working Paper: Regularizing Priors for Linear Inverse Problems (2013)
Working Paper: Regularizing Priors for Linear Inverse Problems (2013)
Working Paper: Regularizing priors for linear inverse problems (2010)
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