Bayesian Fixed-Domain Asymptotics for Covariance Parameters in Spatial Gaussian Process Models

Oregon Convention Center 

Gaussian process models typically contain finite dimensional parameters in the covariance function that need to be estimated from the data. We establish new Bayesian fixed-domain asymptotic theory for the covariance parameters in spatial Gaussian process regression models with an isotropic Matern covariance function, which has many applications in spatial statistics. For the model without nugget, we show that when the domain dimension is less than or equal to three, the microergodic parameter and the range parameter are asymptotically independent in the posterior. While the posterior distribution of microergodic parameter is asymptotically normal with a shrinking variance, the posterior distribution of range parameter does not converge to any point mass in general. For the model with nugget, we derive new evidence lower bound and consistent higher-order quadratic variation estimators, which lead to explicit posterior contraction rates for both the microergodic parameter and the nugget parameter. We further study the asymptotic efficiency of Bayesian kriging prediction. All the new theoretical results are verified in numerical experiments and real data analysis.