2020-08-15T11:07:35Zhttp://tse.rima1.fr/academ-xoai/oai2/puboai:tse-fr.eu:228842015-07-31T18:00:02Zut1tse:publication
Regularizing priors for linear inverse problems
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Florens
Jean-Pierre
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Author
Auteur
Simoni
Anna
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Author
Auteur
2010-05
eng
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TSE Working Paper
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http://tse-fr.eu/pub/7802
10-175
http://tse-fr.eu/pub/22884
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.
http://tse-fr.eu/pub/22884
http://www.tse-fr.eu/sites/default/files/medias/doc/wp/etrie/10-175.pdf
Toulouse School of Economics
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eng
English
anglais