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Rates of Convergence for a Bayesian Level Set Estimation

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  • GHISLAINE GAYRAUD
  • JUDITH ROUSSEAU

Abstract

. We are interested in estimating level sets using a Bayesian non‐parametric approach, from an independent and identically distributed sample drawn from an unknown distribution. Under fairly general conditions on the prior, we provide an upper bound on the rate of convergence of the Bayesian level set estimate, via the rate at which the posterior distribution concentrates around the true level set. We then consider, as an application, the log‐spline prior in the two‐dimensional unit cube. Assuming that the true distribution belongs to a class of Hölder, we provide an upper bound on the rate of convergence of the Bayesian level set estimates. We compare our results with existing rates of convergence in the frequentist non‐parametric literature: the Bayesian level set estimator proves to be competitive and is also easy to compute, which is of no small importance. A simulation study is given as an illustration.

Suggested Citation

  • Ghislaine Gayraud & Judith Rousseau, 2005. "Rates of Convergence for a Bayesian Level Set Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 639-660, December.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:4:p:639-660
    DOI: 10.1111/j.1467-9469.2005.00448.x
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    1. Ghislaine Gayraud & Judith Rousseau, 2002. "Nonparametric Bayesian Estimation of Level Sets," Working Papers 2002-03, Center for Research in Economics and Statistics.
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    Cited by:

    1. Burman, Prabir & Polonik, Wolfgang, 2009. "Multivariate mode hunting: Data analytic tools with measures of significance," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1198-1218, July.
    2. Mammen, Enno & Polonik, Wolfgang, 2013. "Confidence regions for level sets," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 202-214.

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