Regularizing Priors for Linear Inverse Problems

Florens, Jean-Pierre and Simoni, Anna (2016) Regularizing Priors for Linear Inverse Problems. Econometric Theory, 32 (1). pp. 71-121.

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Official URL: http://tse-fr.eu/pub/28974

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 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 such 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.

Item Type: Article
Language: English
Date: February 2016
Refereed: Yes
JEL codes: C11 - Bayesian Analysis
C13 - Estimation
C14 - Semiparametric and Nonparametric Methods
Subjects: B- ECONOMIE ET FINANCE
Divisions: TSE-R (Toulouse)
Site: UT1
Date Deposited: 16 Mar 2015 14:55
Last Modified: 07 Mar 2018 13:23
OAI ID: oai:tse-fr.eu:28974
URI: http://publications.ut-capitole.fr/id/eprint/16687

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