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:空间高斯进程回归模型通常含有需要从数据中估算的有限维共变参数。 我们研究了贝叶西亚对共变参数的估算,包括固定面静态静态静态静态静态反应功能一般类别中的共变参数,由于空间观测中日益强烈的依赖性,这在理论上具有挑战性。 我们提议对Schwartz的一致性理论进行新颖的修改,以显示包括核糖在内的共变参数的后继收缩率。 我们从中得出新的多元证据,并提议一个更低约束的一致的较高级二次变异估计值,用极小尾巴来满足浓度不平等。 我们的贝叶斯固定多位静态静态理论导致在一般分层取样设计下,对等离子磁场变异参数的显性后子收缩率。 我们核查我们的理论以及模拟研究和海面温度数据应用中的巴伊斯预测性表现。