Fixed-Domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Abstract

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. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian inference
• Evidence lower bound
• Higher-order quadratic variation
• Matérn covariance function

Acknowledgments

The authors thank Professor Wei-Liem Loh for helpful discussion. The authors are grateful to the Associate Editor and the reviewers for their valuable and insightful comments that have helped improve the article.

Additional information

Funding

The authors gratefully acknowledge the support of Singapore Ministry of Education Academic Research Funds Tier 1 grant A-0004822-00-00.