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Title: Bayesian fixed-domain asymptotics for covariance parameters in Gaussian process models Authors:  Cheng Li - National University of Singapore (Singapore) [presenting]
Abstract: Gaussian process models are widely used for modeling spatial processes. We focus on the Gaussian process with isotropic Matern covariance functions. Under fixed-domain asymptotics, it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the range (or length-scale) parameter cannot. Motivated by this frequentist result, we prove a Bernstein-von Mises theorem for the covariance parameters under a Bayesian framework. Under the fixed-domain asymptotics, the posterior distribution of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance. In contrast, the posterior of the range parameter does not necessarily converge. We further propose a new property called the posterior asymptotic efficiency in linear prediction, and show that the Bayesian kriging predictor at a new spatial location with covariance parameters randomly drawn from their posterior has the same prediction mean squared error as if the true parameters were known. We illustrate these asymptotic results in numerical examples.