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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.<\/jats:p>","DOI":"10.1017\/s0266466614000796","type":"journal-article","created":{"date-parts":[[2014,11,6]],"date-time":"2014-11-06T04:29:04Z","timestamp":1415248144000},"page":"71-121","source":"Crossref","is-referenced-by-count":16,"title":["REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS"],"prefix":"10.1017","volume":"32","author":[{"given":"Jean-Pierre","family":"Florens","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anna","family":"Simoni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,11,6]]},"reference":[{"key":"S0266466614000796_ref25","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/23\/6\/012"},{"key":"S0266466614000796_ref31","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOS920"},{"key":"S0266466614000796_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.01.032"},{"key":"S0266466614000796_ref39","doi-asserted-by":"publisher","DOI":"10.1007\/BF00533743"},{"key":"S0266466614000796_ref38","doi-asserted-by":"publisher","DOI":"10.1137\/1140017"},{"key":"S0266466614000796_ref43","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOS678"},{"key":"S0266466614000796_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0559-3"},{"key":"S0266466614000796_ref19","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9469.2011.00784.x"},{"key":"S0266466614000796_ref6","doi-asserted-by":"crossref","first-page":"5633","DOI":"10.1016\/S1573-4412(07)06077-1","volume-title":"Handbook of Econometrics","volume":"6","author":"Carrasco","year":"2005"},{"key":"S0266466614000796_ref17","unstructured":"Florens J.P. & Simoni A. 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