Robust Prediction Variance Estimation for Gaussian Process Regression Under Covariance Smoothness Misspecification
This study addresses the underestimation of predictive variance in Gaussian process regression caused by misspecification of the covariance function’s smoothness, which induces bias in the mean squared prediction error (MSPE) estimation of the empirical best linear unbiased predictor (EBLUP). The authors establish, for the first time, that when the measures induced by the true and assumed covariance functions are mutually singular, the MSPE bias converges to a strictly positive limit that varies smoothly with the prediction location. Building on this insight, they propose a novel robust MSPE estimator that explicitly accounts for covariance uncertainty. Both theoretical analysis and numerical experiments demonstrate that the proposed estimator substantially outperforms four state-of-the-art alternatives across various smoothness misspecification scenarios, with its advantage becoming more pronounced as the degree of misspecification increases.