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Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models

Author

Listed:
  • Dr. Prof. Jan Johannes
  • Dr. Anna Simoni

    (CNRS - Centre National de la Recherche Scientifique, 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)

  • Dr. Schenk

Abstract

In an indirect Gaussian sequence space model we derive lower and upper bounds for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value THETA° that generates the data. While this establishes posterior consistency, the concentration rate depends on both THETA° and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each THETA° separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive for mildly ill-posed inverse problems. This means that, given a parameter THETA° or a class of parameters, the posterior distribution contracts at the oracle rate or at the minimax rate over the class, respectively. Notably, the hierarchical prior does not depend neither on THETA° nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.
(This abstract was borrowed from another version of this item.)
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Dr. Prof. Jan Johannes & Dr. Anna Simoni & Dr. Schenk, 2020. "Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models," Post-Print hal-02903256, HAL.
    • Handle: RePEc:hal:journl:hal-02903256
    DOI: 10.15609/annaeconstat2009.137.0083
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    References listed on IDEAS

    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. repec:dau:papers:123456789/11426 is not listed on IDEAS
    3. Johannes, Jan & Schwarz, Maik, 2013. "Adaptive Gaussian Inverse Regression with Partially Unknown Operator," LIDAM Reprints ISBA 2013022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Julyan Arbel & Ghislaine Gayraud & Judith Rousseau, 2013. "Bayesian Optimal Adaptive Estimation Using a Sieve Prior," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 549-570, September.
    5. Julyan Arbel & Ghislaine Gayraud & Judith Rousseau, 2013. "Bayesian Optimal Adaptive Estimation Using a Sieve prior," Working Papers 2013-19, Center for Research in Economics and Statistics.
    6. Comte, Fabienne & Johannes, Jan, 2012. "Adaptive functional linear regression," LIDAM Reprints ISBA 2012031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. F. Abramovich & T. Sapatinas & B. W. Silverman, 1998. "Wavelet thresholding via a Bayesian approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 725-749.
    8. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
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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
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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