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

A.W. Vaart, van der, J.H. Zanten, van

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118 Citations (Scopus)
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Abstract

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.
Original languageEnglish
Pages (from-to)2655-2675
JournalThe Annals of Statistics
Volume37
Issue number5B
DOIs
Publication statusPublished - 2009

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