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Regularizing Priors For Linear Inverse Problems

Author

Listed:
  • Jean-Pierre Florens
  • Anna Simoni

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique, CNRS - Centre National de la Recherche Scientifique)

Abstract
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.
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Suggested Citation

  • Jean-Pierre Florens & Anna Simoni, 2016. "Regularizing Priors For Linear Inverse Problems," Post-Print hal-03089887, HAL.
    • Handle: RePEc:hal:journl:hal-03089887
    DOI: 10.1017/S0266466614000796
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    as
    1. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
    2. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(3), pages 497-521, June.
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    5. Florens, Jean-Pierre & Sbaï, Erwann, 2010. "Local Identification In Empirical Games Of Incomplete Information," Econometric Theory, Cambridge University Press, vol. 26(6), pages 1638-1662, December.
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    11. Florens, Jean-Pierre & Johannes, Jan & Van Bellegem, Sebastien, 2011. "Identification and estimation by penalization in Nonparametric Instrumental Regression," LIDAM Reprints ISBA 2011046, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    22. Jean-Pierre Florens & Anna Simoni, 2012. "Regularized Posteriors in Linear Ill-Posed Inverse Problems," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(2), pages 214-235, June.
    23. Joel L. Horowitz, 2013. "Adaptive nonparametric instrumental variables estimation: empirical choice of the regularization parameter," CeMMAP working papers CWP30/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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    Cited by:

    1. Siddhartha Chib & Minchul Shin & Anna Simoni, 2022. "Bayesian estimation and comparison of conditional moment models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 740-764, July.
    2. Jean-Pierre Florens & Anna Simoni, 2021. "Gaussian Processes and Bayesian Moment Estimation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(2), pages 482-492, March.
    3. Laurent Ferrara & Anna Simoni, 2023. "When are Google Data Useful to Nowcast GDP? An Approach via Preselection and Shrinkage," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 41(4), pages 1188-1202, October.
    4. Florens, Jean-Pierre & Simoni, Anna, 2016. "Regularizing Priors For Linear Inverse Problems," Econometric Theory, Cambridge University Press, vol. 32(1), pages 71-121, February.
    5. Florens, Jean-Pierre & Simoni, Anna, 2012. "Nonparametric estimation of an instrumental regression: A quasi-Bayesian approach based on regularized posterior," Journal of Econometrics, Elsevier, vol. 170(2), pages 458-475.
    6. Jan Johannes & Anna Simoni & Rudolf Schenk, 2020. "Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models," Annals of Economics and Statistics, GENES, issue 137, pages 83-116.
    7. Liao, Yuan & Jiang, Wenxin, 2011. "Posterior consistency of nonparametric conditional moment restricted models," MPRA Paper 38700, University Library of Munich, Germany.

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    More about this item

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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