Gaussian process regression in its most simplified form assumes normal homoscedastic noise and utilizes analytically tractable mean and covariance functions of predictive posterior distribution using Gaussian conditioning. Its hyperparameters are estimated by maximizing the evidence, commonly known as type II maximum likelihood estimation. Unfortunately, Bayesian inference based on Gaussian likelihood is not robust to outliers, which are often present in the observational training data sets. To overcome this problem, we propose a robust process model in the Gaussian process framework with the likelihood of observed data expressed as the Huber probability distribution. The proposed model employs weights based on projection statistics to scale residuals and bound the influence of vertical outliers and bad leverage points on the latent functions estimates while exhibiting a high statistical efficiency at the Gaussian and thick tailed noise distributions. The proposed method is demonstrated by two real world problems and two numerical examples using datasets with additive errors following thick tailed distributions such as Students t, Laplace, and Cauchy distribution.