# Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions

arXiv: https://arxiv.org/abs/2001.10965 · HTML: https://mlsanity.com/paper/2001.10965/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2020.

## Built on by (most cited first, 15 of 15)

- [Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed](https://mlsanity.com/paper/2203.09179/) (2022)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Small Sample Spaces for Gaussian Processes](https://mlsanity.com/paper/2103.03169/) (2021)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Gaussian process interpolation with conformal prediction: methods and comparative analysis](https://mlsanity.com/paper/2407.08271/) (2024)
- [Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation](https://mlsanity.com/paper/2203.05400/) (2022)
- [Linear-Time Probabilistic Solutions of Boundary Value Problems](https://mlsanity.com/paper/2106.07761/) (2021)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Active Discrimination Learning for Gaussian Process Models](https://mlsanity.com/paper/2211.11624/) (2022)
- [Interpolation with the polynomial kernels](https://mlsanity.com/paper/2212.07658/) (2022)
- [From Target Tracking to Targeting Track -- Part III: Stochastic Process Modeling and Online Learning](https://mlsanity.com/paper/2503.05799/) (2025)
- [Weighted Leave-One-Out Cross Validation](https://mlsanity.com/paper/2505.19737/) (2025)
- [Why not to use the Gaussian kernel](https://mlsanity.com/paper/2608.26974/) (2026)

## Abstract (arXiv)

Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the dataset. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless dataset. Specifically, we consider the scenario where the scale parameter of a Sobolev kernel (such as a Mat'ern kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model in the sense that the model can become "slowly" overconfident at worst, regardless of the difference between the smoothness of the data-generating function and that expected by the model. The analysis is based on a combination of techniques from nonparametric regression and scattered data interpolation. Empirical results are provided in support of the theoretical findings.