Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth

Details Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth

2009-08-25

arxiv.org/abs/0908.3556v1

Annals of Statistics 2009, Vol. 37, No. 5B, 2655-2675

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-AOS678 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

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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