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Simultaneous local and global adaptivity of Bayesian wavelet estimators in nonparametric regression

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If you have a question about this talk, please contact Richard Nickl .

We consider Bayesian wavelet estimators in the context of nonparametric regression. The most commonly used wavelet estimators are separable, i.e. estimator of each wavelet coefficient is based only on its own observation. However, as it was shown by Cai (2008) for the white noise model, adaptive separable estimators cannot achieve minimax-optimal rate in L_2 norm (global rate) without paying the price for adaptivity. The nonseparable estimator of Johnstone and Silverman (2005) that uses maximum marginal likelihood approach to estimate some of the parameters, and achieves the optimal global rate, in fact can be interpreted as a Bayesian estimator. We show that it also achieves adaptive minimax-optimal local rate, and discuss other Bayesian wavelet estimators that pool information together across wavelet coefficients.

Bayesian wavelet modelling is usually done in the domain of wavelet coefficients. We discuss how a priori assumptions in wavelet domain transfer to the function domain for the considered estimators. Part of this work is joint with T.Sapatinas (University of Cyprus).

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