期刊
ANNALS OF STATISTICS
卷 37, 期 5B, 页码 2655-2675出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/08-AOS678
关键词
Rate of convergence; posterior distribution; adaptation; Bayesian inference; nonparametric density estimation; nonparametric regression; classification; Gaussian process priors
资金
- Netherlands Organization for Scientific Research NWO
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 all 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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