RATES OF CONTRACTION FOR POSTERIOR DISTRIBUTIONS IN Lr-METRICS, 1 ≤ r ≤ ∞

• 成果类型: Article
• 署名作者: Gine, Evarist; Nickl, Richard
• 署名单位: University of Connecticut; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS924
• 发表日期: 2011
• 页码: 2883-2911
• 关键词: CONVERGENCE-RATES; Concentration inequalities; dirichlet mixtures; maximum-likelihood; Consistency; THEOREMS; SPACES

摘要

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking L-r-norm neighborhoods, 1 <= r <= infinity, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for 1 <= r <= 2, but deteriorate as r increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all L-r-norms, 1 <= r <= infinity.