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Abstract:
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Gaussian process models typically contain finite dimensional parameters in the covariance functions that need to be estimated from the data. Under the fixed-domain asymptotics framework, we establish a Bernstein-von Mises theorem for the covariance parameters in a Gaussian process with an isotropic Matern covariance function when the dimension of data is less than or equal to three, and show that the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors asymptotically. The posterior of the microergodic parameter converges to a normal distribution with a shrinking variance, while the posterior of the range parameter does not necessarily converge to any degenerate distribution in general. We further show that the Bayesian prediction with covariance parameters randomly drawn from their posterior distribution satisfies the asymptotic efficiency in linear prediction. We verify these asymptotic results in numerical examples.
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