ADAPTIVE BAYESIAN ESTIMATION USING A GAUSSIAN RANDOM FIELD WITH INVERSE GAMMA BANDWIDTH

• 成果类型: Article
• 署名作者: van der Vaart, A. W.; van Zanten, J. H.
• 署名单位: Vrije Universiteit Amsterdam
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS678
• 发表日期: 2009
• 页码: 2655-2675
• 关键词: nonparametric binary regression; DENSITY-ESTIMATION; convergence-rates; posterior distributions; STOCHASTIC-PROCESSES; model selection; Metric Entropy; process priors; Consistency; INEQUALITY

摘要

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.