# Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation

arXiv: https://arxiv.org/abs/2203.05400 · HTML: https://mlsanity.com/paper/2203.05400/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 67 references that are in the map.
- First posted on arXiv 10 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On the marginal likelihood and cross-validation](https://mlsanity.com/paper/1905.08737/) (2019)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions](https://mlsanity.com/paper/2001.10965/) (2020)
- [Small Sample Spaces for Gaussian Processes](https://mlsanity.com/paper/2103.03169/) (2021)

## Built on by (most cited first, 2 of 2)

- [A class of modular and flexible covariate-based covariance functions for nonstationary spatial modeling](https://mlsanity.com/paper/2410.16716/) (2024)
- [Active Discrimination Learning for Gaussian Process Models](https://mlsanity.com/paper/2211.11624/) (2022)

## Abstract (arXiv)

It is common to model a deterministic response function, such as the output of a computer experiment, as a Gaussian process with a Mat'ern covariance kernel. The smoothness parameter of a Mat'ern kernel determines many important properties of the model in the large data limit, including the rate of convergence of the conditional mean to the response function. We prove that the maximum likelihood estimate of the smoothness parameter cannot asymptotically undersmooth the truth when the data are obtained on a fixed bounded subset of ℝd. That is, if the data-generating response function has Sobolev smoothness nu0 > d/2, then the smoothness parameter estimate cannot be asymptotically less than nu0. The lower bound is sharp. Additionally, we show that maximum likelihood estimation recovers the true smoothness for a class of compactly supported self-similar functions. For cross-validation we prove an asymptotic lower bound nu0 - d/2, which however is unlikely to be sharp. The results are based on approximation theory in Sobolev spaces and some general theorems that restrict the set of values that the parameter estimators can take.