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Bayesian Optimal Adaptive Estimation Using a Sieve prior

Author

Listed:
  • Julyan Arbel

    () (CREST)

  • Ghislaine Gayraud

    () (CREST)

  • Judith Rousseau

    () (CREST)

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.

Suggested Citation

  • 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.
  • Handle: RePEc:crs:wpaper:2013-19
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    File URL: http://crest.science/RePEc/wpstorage/2013-19.pdf
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    References listed on IDEAS

    as
    1. Felix Abramovich & Claudia Angelini & Daniela Canditiis, 2007. "Pointwise optimality of Bayesian wavelet estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 425-434, September.
    2. 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.
    3. repec:dau:papers:123456789/1908 is not listed on IDEAS
    4. repec:dau:papers:123456789/7335 is not listed on IDEAS
    5. 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.
    6. 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, Center for Research in Economics and Statistics.
    7. repec:dau:papers:123456789/3984 is not listed on IDEAS
    8. repec:dau:papers:123456789/4659 is not listed on IDEAS
    9. 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.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. repec:eee:stapro:v:130:y:2017:i:c:p:5-11 is not listed on IDEAS
    2. van Waaij, Jan & van Zanten, Harry, 2017. "Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 93-99.
    3. Weining Shen & Subhashis Ghosal, 2015. "Adaptive Bayesian Procedures Using Random Series Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1194-1213, December.

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