Bayesian Fixed-domain Asymptotics for Covariance Parameters in a Gaussian Process Model

TL;DR

This paper investigates Bayesian asymptotic behavior of covariance parameters in Gaussian process models with isotropic Matern covariance, revealing independence of certain posteriors and convergence properties in fixed-domain settings, with applications in spatial statistics.

Contribution

It provides new theoretical insights into the asymptotic independence and convergence of covariance parameter posteriors in Gaussian processes under fixed-domain asymptotics, including explicit results for the Ornstein-Uhlenbeck case.

Findings

Posterior of the microergodic parameter converges to a normal distribution.

Posterior of the range parameter does not generally concentrate at a point.

The joint posterior factors into independent marginals for certain parameters.

Abstract

Gaussian process models typically contain finite dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matern covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not converge to any point mass distribution in general. Our theory allows an unbounded prior support for the range parameter and flexible designs of sampling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covariance parameters randomly drawn from their posterior distribution. In the special case of one-dimensional Ornstein-Uhlenbeck process, we derive explicitly the limiting posterior of the range parameter and the posterior convergence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.