Improved Most Likely Heteroscedastic Gaussian Process Regression via Bayesian Residual Moment Estimator

This paper proposes an improved most likely heteroscedastic Gaussian process (MLHGP) algorithm to handle a kind of nonlinear regression problems involving input-dependent noise. The improved MLHGP follows the same learning scheme as the current algorithm by use of two Gaussian processes (GPs), with the first GP for recovering the unknown function and the second GP for modeling the input-dependent noise. Unlike the current MLHGP pursuing an empirical estimate of the noise level which is provably biased in most of local noise cases, the improved algorithm gives rise to an approximately unbiased estimate of the input-dependent noise. The approximately unbiased noise estimate is elicited from Bayesian residuals by the method of moments. As a by-product of this improvement, the expectation maximization (EM)-like procedure in the current MLHGP is avoided such that the improved algorithm requires only standard GP learnings to be performed twice. Four benchmark experiments, consisting of two synthetic cases and two real-world datasets, demonstrate that the improved MLHGP algorithm outperforms the current version not only in accuracy and stability, but also in computational efficiency.