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Regularizing priors for linear inverse problems

Florens, Jean-Pierre and Simoni, Anna (2010) Regularizing priors for linear inverse problems. TSE Working Paper, n. 10-175

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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.

Item Type: Monograph (Working Paper)
Language: English
Date: May 2010
Divisions: TSE-R (Toulouse)
Site: UT1
Date Deposited: 18 Jan 2012 06:02
Last Modified: 07 Mar 2018 13:22
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