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April 2000 Bayesian aspects of some nonparametric problems
Linda H. Zhao
Ann. Statist. 28(2): 532-552 (April 2000). DOI: 10.1214/aos/1016218229


We study the Bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation. We consider the asymptotic properties of Bayes procedures for conjugate (= Gaussian) priors.

We show that so long as the prior puts nonzero measure on the very large parameter set of interest then the Bayes estimators are not satisfactory. More specifically, we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space. Thus all Bayes estimators for proper Gaussian priors have zero asymptotic efficiency in this minimax sense.

We then present a class of priors whose Bayes procedures attain the optimal minimax rate of convergence. These priors may be viewed as compound, or hierarchical, mixtures of suitable Gaussian distributions.


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Linda H. Zhao. "Bayesian aspects of some nonparametric problems." Ann. Statist. 28 (2) 532 - 552, April 2000.


Published: April 2000
First available in Project Euclid: 15 March 2002

zbMATH: 1010.62025
MathSciNet: MR1790008
Digital Object Identifier: 10.1214/aos/1016218229

Primary: 62G07
Secondary: 62A15 , 62G20

Keywords: Bayes , conjugate priors , minimax , Nonparametric regression , White noise

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 2 • April 2000
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