Open Access
October 2009 Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
A. W. van der Vaart, J. H. van Zanten
Ann. Statist. 37(5B): 2655-2675 (October 2009). DOI: 10.1214/08-AOS678


We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, whatever the regularity level of the data-generating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive.


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A. W. van der Vaart. J. H. van Zanten. "Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth." Ann. Statist. 37 (5B) 2655 - 2675, October 2009.


Published: October 2009
First available in Project Euclid: 17 July 2009

zbMATH: 1173.62021
MathSciNet: MR2541442
Digital Object Identifier: 10.1214/08-AOS678

Primary: 62-07 , 62H30
Secondary: 65U05 , 68T05

Keywords: Adaptation , Bayesian inference , ‎classification‎ , Gaussian process priors , Nonparametric density estimation , Nonparametric regression , posterior distribution , rate of convergence

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5B • October 2009
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