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

The Fall 2022 SDS Seminar Series starts on Friday, September 9th from 2:00 p.m. to 3:00 p.m. with Dr. Cheng Li (Assistant Professor in Department of Statistics and Data Science at National University of Singapore). This event is virtual.

Title: Bayesian Fixed-domain Asymptotics for Covariance Parameters in Spatial Gaussian Process Models

Abstract: Gaussian process models are widely used in Bayesian inference for spatially referenced data. We discuss some recent advances in the theory of Bayesian spatial Gaussian process regression models without and with the nugget effect. We mainly study the Bayesian estimation of parameters in the Gaussian process covariance function under the fixed-domain asymptotics regime, as well as its implications on the Bayesian posterior prediction. For the model without nugget, also known as universal kriging, we derive the limiting joint posterior distribution of the microergodic parameter and the range parameter. We further show that the Bayesian kriging predictor satisfies the posterior asymptotic efficiency in linear prediction. For the more challenging model with a nugget, we propose a new framework to derive the Bayesian posterior contraction rates for the microergodic parameter and the nugget. We illustrate these theoretical results with numerical experiments and real data analysis.