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 Matern 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.

翻译：空间高斯过程回归模型通常包含需从数据中估计的有限维协方差参数。本文研究固定域渐近框架下，一类广义平稳协方差函数中协方差参数（包括金块参数）的贝叶斯估计问题。由于空间观测值间的强依赖性，该问题在理论上具有挑战性。我们提出一种改进的施瓦茨相容性定理，用于证明含金块参数的协方差参数后验收缩率。通过推导多项式证据下界，并构造满足指数小尾浓度不等式的高阶二次变分一致估计量，建立了贝叶斯固定域渐近理论。该理论给出了各向同性马特恩协方差函数中微遍历参数与金块参数在广义分层抽样设计下的显式后验收缩率。通过数值模拟研究及海表温度数据应用，验证了理论结果与贝叶斯预测性能。