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.
Publié le : 2009-10-15
Classification:
Rate of convergence,
posterior distribution,
adaptation,
Bayesian inference,
nonparametric density estimation,
nonparametric regression,
classification,
Gaussian process priors,
62H30,
62-07,
65U05,
68T05
@article{1247836664,
author = {van der Vaart, A. W. and van Zanten, J. H.},
title = {Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth},
journal = {Ann. Statist.},
volume = {37},
number = {1},
year = {2009},
pages = { 2655-2675},
language = {en},
url = {http://dml.mathdoc.fr/item/1247836664}
}
van der Vaart, A. W.; van Zanten, J. H. Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. Ann. Statist., Tome 37 (2009) no. 1, pp. 2655-2675. http://gdmltest.u-ga.fr/item/1247836664/