Adaptive Bayesian Nonparametric Regression Using a Kernel Mixture of Polynomials with Application to Partial Linear Models

作     者：Xie, Fangzheng Xu, Yanxun 

作者机构：Johns Hopkins Univ Dept Appl Math & Stat Baltimore MD 21218 USA 

出 版 物：《BAYESIAN ANALYSIS》 (Bayesian Anal.)

年 卷 期：2020年第15卷第1期

页      面：159-186页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

主　　题：Bayesian nonparametric regression Bernstein-von Mises limit metric entropies partial linear model rate of contraction 

摘      要：We propose a kernel mixture of polynomials prior for Bayesian nonparametric regression. The regression function is modeled by local averages of polynomials with kernel mixture weights. We obtain the minimax-optimal contraction rate of the full posterior distribution up to a logarithmic factor by estimating metric entropies of certain function classes. Under the assumption that the degree of the polynomials is larger than the unknown smoothness level of the true function, the posterior contraction behavior can adapt to this smoothness level provided an upper bound is known. We also provide a frequentist sieve maximum likelihood estimator with a near-optimal convergence rate. We further investigate the application of the kernel mixture of polynomials to partial linear models and obtain both the near-optimal rate of contraction for the nonparametric component and the Bernstein-von Mises limit (i.e., asymptotic normality) of the parametric component. The proposed method is illustrated with numerical examples and shows superior performance in terms of computational efficiency, accuracy, and uncertainty quantification compared to the local polynomial regression, DiceKriging, and the robust Gaussian stochastic process.