The Annals of Statistics
- Ann. Statist.
- Volume 37, Number 5B (2009), 2655-2675.
Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
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.
Ann. Statist., Volume 37, Number 5B (2009), 2655-2675.
First available in Project Euclid: 17 July 2009
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van der Vaart, A. W.; van Zanten, J. H. Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. Ann. Statist. 37 (2009), no. 5B, 2655--2675. doi:10.1214/08-AOS678. https://projecteuclid.org/euclid.aos/1247836664