  Probabilistic regression models typically use the Maximum Likelihood
Estimation or Cross-Validation to fit parameters. These methods can give an
advantage to the solutions that fit observations on average, but they do not
pay attention to the coverage and the width of Prediction Intervals. A robust
two-step approach is used to address the problem of adjusting and calibrating
Prediction Intervals for Gaussian Processes Regression. First, the covariance
hyperparameters are determined by a standard Cross-Validation or Maximum
Likelihood Estimation method. A Leave-One-Out Coverage Probability is
introduced as a metric to adjust the covariance hyperparameters and assess the
optimal type II Coverage Probability to a nominal level. Then a relaxation
method is applied to choose the hyperparameters that minimize the Wasserstein
distance between the Gaussian distribution with the initial hyperparameters
(obtained by Cross-Validation or Maximum Likelihood Estimation) and the
proposed Gaussian distribution with the hyperparameters that achieve the
desired Coverage Probability. The method gives Prediction Intervals with
appropriate coverage probabilities and small widths.
