# Robust Gaussian Process Regression with Huber Likelihood

arXiv: https://arxiv.org/abs/2301.07858 · HTML: https://mlsanity.com/paper/2301.07858/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 34 references that are in the map.
- First posted on arXiv 19 Jan 2023.

## Builds on (most cited first, 3 of 3)

- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Laplace approximation and the natural gradient for Gaussian process regression with the heteroscedastic Student-t model](https://mlsanity.com/paper/1712.07437/) (2017)
- [Bayesian Robustness to Outliers in Linear Regression and Ratio Estimation](https://mlsanity.com/paper/1612.05307/) (2016)

## Built on by (most cited first, 6 of 6)

- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Robust Gaussian Processes via Relevance Pursuit](https://mlsanity.com/paper/2410.24222/) (2024)
- [Deep Random Features for Scalable Interpolation of Spatiotemporal Data](https://mlsanity.com/paper/2412.11350/) (2024)
- [Robust and Computation-Aware Gaussian Processes](https://mlsanity.com/paper/2505.21133/) (2025)
- [Sparse Variational Contaminated Noise Gaussian Process Regression with Applications in Geomagnetic Perturbations Forecasting](https://mlsanity.com/paper/2402.17570/) (2024)
- [Joint Model and Data Sparsification via the Marginal Likelihood](https://mlsanity.com/paper/2605.29908/) (2026)

## Abstract (arXiv)

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.