 Solution

Gaussian process regression supplies a predictive distribution and uncertainty estimates. A Gaussian process regression posterior gives both a posterior mean and a posterior variance, allowing credible intervals for the latent function and a posterior predictive distribution for future noisy observations. The kernel ridge regression estimator by itself specifies a point estimate.

With the same positive-semidefinite kernel and matching regularization, the Gaussian process regression posterior mean equals the kernel ridge regression prediction. Thus the main additional benefit is the explicit probability distribution of predictions, conditional on the chosen Gaussian process model and its hyperparameters, rather than an automatic improvement in point-prediction accuracy.