Robust Prediction Interval estimation for Gaussian Processes by Cross-Validation method

Probabilistic regression models typically use the Maximum Likelihood Estimation or Cross-Validation to fit parameters. Unfortunately, these methods may give advantage to the solutions that fit observations in average, but they do not pay attention to the coverage and the width of Prediction Intervals. In this paper, we address the question of adjusting and calibrating Prediction Intervals for Gaussian Processes Regression. First we determine the model's parameters by a standard Cross-Validation or Maximum Likelihood Estimation method then we adjust the parameters to assess the optimal type II Coverage Probability to a nominal level. We apply a relaxation method to choose parameters that minimize the Wasserstein distance between the Gaussian distribution of the initial parameters (Cross-Validation or Maximum Likelihood Estimation) and the proposed Gaussian distribution among the set of parameters that achieved the desired Coverage Probability.