BAYESIAN FIXED-DOMAIN ASYMPTOTICS FOR COVARIANCE PARAMETERS IN A GAUSSIAN PROCESS MODEL

• 成果类型: Article
• 署名作者: LI, Cheng
• 署名单位: National University of Singapore
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2230
• 发表日期: 2022
• 页码: 3334-3363
• 关键词: maximum-likelihood-estimation; stochastic-process model; random-field; linear predictions; Nonparametric Regression; data sets; inference; approximation; optimality; EFFICIENCY

摘要

Gaussian process models typically contain finite-dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matern covariance function, which has many applications in spatial statistics. We show that when the dimen-sion of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored in-dependently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking vari-ance, while the posterior distribution of the range parameter does not con-verge to any point mass distribution in general. Our theory allows an un-bounded prior support for the range parameter and flexible designs of sam-pling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covari-ance parameters randomly drawn from their posterior distribution. In the spe-cial case of one-dimensional Ornstein-Uhlenbeck process, we derive explic-itly the limiting posterior of the range parameter and the posterior conver-gence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.