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Bayesian Optimal Adaptive Estimation Using a Sieve prior

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  • Julyan Arbel

    ()
    (CREST)

  • Ghislaine Gayraud

    (CREST)

  • Judith Rousseau

    (CREST)

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    Abstract

    We derive rates of contraction of posterior distributions on nonparametric models resulting from sieve priors. The aim of the paper is to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter space is, e.g., a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate.

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    Bibliographic Info

    Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 2013-19.

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    Length: 32
    Date of creation: Dec 2013
    Date of revision:
    Handle: RePEc:crs:wpaper:2013-19

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    Related research

    Keywords: adaptation; minimax criteria; nonparametric models; rate of contraction; sieve prior; white noise model;

    References

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    1. Judith Rousseau & Nicolas Chopin & Brunero Liseo, 2010. "Bayesian Nonparametric Estimation of the Spectral Density of a Long or Intermediate Memory Gaussian Process," Working Papers 2010-38, Centre de Recherche en Economie et Statistique.
    2. Rivoirard, Vincent & Rousseau, Judith, 2012. "Posterior concentration rates for infinite dimensional exponential families," Economics Papers from University Paris Dauphine 123456789/7335, Paris Dauphine University.
    3. Felix Abramovich & Umberto Amato & Claudia Angelini, 2004. "On Optimality of Bayesian Wavelet Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 31(2), pages 217-234.
    4. Rousseau, Judith, 2010. "Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density," Economics Papers from University Paris Dauphine 123456789/3984, Paris Dauphine University.
    5. Robert, Christian P., 2007. "The Bayesian Choice: From Decision Theoretic Foundations to Computational Implementation," Economics Papers from University Paris Dauphine 123456789/1908, Paris Dauphine University.
    6. Felix Abramovich & Vadim Grinshtein & Athanasia Petsa & Theofanis Sapatinas, 2010. "On Bayesian testimation and its application to wavelet thresholding," Biometrika, Biometrika Trust, vol. 97(1), pages 181-198.
    7. Babenko, Alexandra & Belitser, Eduard, 2009. "On the posterior pointwise convergence rate of a Gaussian signal under a conjugate prior," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 670-675, March.
    8. Felix Abramovich & Claudia Angelini & Daniela Canditiis, 2007. "Pointwise optimality of Bayesian wavelet estimators," Annals of the Institute of Statistical Mathematics, Springer, vol. 59(3), pages 425-434, September.
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