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The Gibbs reference posterior distribution provides an objectivefull-Bayesian solution to the problem of prediction of a stationary Gaussianprocess with Mat\'ern anisotropic kernel. A full-Bayesian approach is possible,because the posterior distribution is expressed as the invariant distributionof a uniformly ergodic Markovian kernel for which we give an explicitexpression. In this paper, we show that it is appropriate for the UniversalKriging framework, that is when an unknown function is added to the stationaryGaussian process. We give sufficient conditions for the existence and proprietyof the Gibbs reference posterior that apply to a wide variety of practicalcases and illustrate the method with several examples. Finally, simulations ofGaussian processes suggest that the Gibbs reference posterior has goodfrequentist properties in terms of coverage of prediction intervals.

A Comprehensive Bayesian Treatment of the Universal Kriging model with Mat\'ern correlation kernels