An empirical evaluation of first-order Noisy Input Gaussian Process Regression under increasing input uncertainty

Gaussian Process (GP) regression provides a flexible probabilistic framework for learning non-linear input–output relationships, but standard formulations typically assume that input variables are observed without error. Noisy Input Gaussian Process Regression (NIGP) addresses this limitation by propagating input uncertainty through a first-order Taylor approximation of the GP posterior mean, thereby introducing an input-dependent effective variance. This study presents a systematic empirical evaluation of the established first-order NIGP formulation under progressively increasing input uncertainty rather than proposing a new mathematical correction. Synthetic experiments were conducted using a sine–trend function with input-noise standard deviations ranging from 0.1 to 1.0 and 50 paired Monte Carlo replicates per condition. Performance was assessed using Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Negative Log Predictive Density (NLPD), Prediction Interval Coverage Probability (PICP), absolute coverage error (ACE), mean prediction interval width (MPIW), Winkler score, and Continuous Ranked Probability Score (CRPS). The first-order approximation was additionally evaluated against Monte Carlo uncertainty propagation and a second-order Taylor reference, while further experiments examined multiple non-linear benchmarks, heterogeneous input uncertainty, and multidimensional correlated input noise. In the principal experiment, GP and NIGP exhibited similar point-prediction performance at low input-noise levels, whereas NIGP produced substantially lower RMSE and MAE from σ x = 0.4 onward, with the largest differences occurring at high noise levels. At σ x = 1.0, mean RMSE decreased from 2.10461 for GP to 0.35937 for NIGP, while mean MAE decreased from 1.59203 to 0.29728. The corresponding bias–variance analysis showed that these improvements were associated with reductions in both predictive bias and variance. Probabilistic performance was noise-dependent: the models were broadly similar at low noise levels, while NIGP produced substantially narrower intervals and lower NLPD, Winkler, and CRPS values at higher noise levels. However, its coverage relative to the nominal 95% level was not uniformly improved. The first-order approximation also deteriorated as input noise increased, with mean relative error rising from approximately 2.1% at σ x = 0.1 to 75.8% at σ x = 1.0, while empirical convergence of the iterative procedure declined from 100% to 34%. The results demonstrate that the empirical benefits of NIGP become increasingly evident under moderate-to-high input uncertainty, but its effectiveness remains constrained by the accuracy of the first-order approximation and the numerical stability of the iterative uncertainty-propagation procedure.