Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget
研究了固定域渐近下含块金参数的协方差函数贝叶斯估计，推导了后验收缩率，并在Matérn协方差函数和分层抽样设计中验证了理论，对空间统计学者判断是否阅读原文有参考价值。
Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz’s consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data.