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Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models

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  • Jan Johannes
  • Anna Simoni
  • Rudolf 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.

Suggested Citation

  • 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.
  • Handle: RePEc:adr:anecst:y:2020:i:137:p:83-116
    DOI: 10.15609/annaeconstat2009.137.0083
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    File URL: https://www.jstor.org/stable/10.15609/annaeconstat2009.137.0083
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    References listed on IDEAS

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    7. 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).
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    More about this item

    Keywords

    Bayesian Nonparametrics; Sieve Prior; Hierarchical Bayes; Exact Concentration Rates; Oracle Optimality; Minimax Theory; Adaptation.;
    All these keywords.

    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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