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

TL;DR

This paper develops a Bayesian framework for estimating covariance parameters, including the nugget, in spatial Gaussian process models under fixed-domain asymptotics, providing explicit contraction rates and validating through simulations and real data.

Contribution

It introduces a novel adaptation of Schwartz's theorem, new evidence lower bounds, and consistent estimators for covariance parameters in spatial Gaussian processes with fixed-domain asymptotics.

Findings

Derived explicit posterior contraction rates for covariance parameters.

Proposed consistent higher-order quadratic variation estimators.

Validated theory and predictive performance through simulations and sea surface temperature data.

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 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.