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Regularizing Priors for Linear Inverse Problems

Jean-Pierre Florens () and Anna Simoni
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Jean-Pierre Florens: GREMAQ - Groupe de recherche en économie mathématique et quantitative - CNRS - Centre National de la Recherche Scientifique - EHESS - École des hautes études en sciences sociales - INRA - Institut National de la Recherche Agronomique - UT1 - Université Toulouse 1 Capitole

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Abstract: This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior distribution we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions so that the posterior mean, in a conjugate- Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

Keywords: nonparametric estimation; Bayesian inverse problems; Gaussian processes; posterior consistency; data-driven method (search for similar items in EconPapers)
Date: 2013-10-16
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00873180v2
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Related works:
Journal Article: REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS (2016) Downloads
Working Paper: Regularizing Priors for Linear Inverse Problems (2013) Downloads
Working Paper: Regularizing Priors for Linear Inverse Problems (2013) Downloads
Working Paper: Regularizing Priors for Linear Inverse Problems (2013)
Working Paper: Regularizing priors for linear inverse problems (2010) Downloads
Working Paper: Regularizing priors for linear inverse problems (2010) Downloads
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