# Uncertainty quantification using martingales for misspecified Gaussian processes

arXiv: https://arxiv.org/abs/2006.07368 · HTML: https://mlsanity.com/paper/2006.07368/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 12 of 12)

- [Misspecified Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2111.05008/) (2021)
- [A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits](https://mlsanity.com/paper/2407.13977/) (2024)
- [Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures](https://mlsanity.com/paper/2309.14298/) (2023)
- [Likelihood Ratio Confidence Sets for Sequential Decision Making](https://mlsanity.com/paper/2311.04402/) (2023)
- [Tighter Confidence Bounds for Sequential Kernel Regression](https://mlsanity.com/paper/2403.12732/) (2024)
- [Early-Exit Neural Networks with Nested Prediction Sets](https://mlsanity.com/paper/2311.05931/) (2023)
- [Adaptation to Misspecified Kernel Regularity in Kernelised Bandits](https://mlsanity.com/paper/2304.13830/) (2023)
- [Meta-Analysis with Untrusted Data](https://mlsanity.com/paper/2407.09387/) (2024)
- [Uncertainty quantification and exploration-exploitation trade-off in humans](https://mlsanity.com/paper/2102.07647/) (2021)
- [Robust Bayesian Optimization via Localized Online Conformal Prediction](https://mlsanity.com/paper/2411.17387/) (2024)
- [ORTHOBO: Orthogonal Bayesian Hyperparameter Optimization](https://mlsanity.com/paper/2605.06454/) (2026)
- [Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization](https://mlsanity.com/paper/2605.20145/) (2026)

## Abstract (arXiv)

We address uncertainty quantification for Gaussian processes (GPs) under misspecified priors, with an eye towards Bayesian Optimization (BO). GPs are widely used in BO because they easily enable exploration based on posterior uncertainty bands. However, this convenience comes at the cost of robustness: a typical function encountered in practice is unlikely to have been drawn from the data scientist's prior, in which case uncertainty estimates can be misleading, and the resulting exploration can be suboptimal. We present a frequentist approach to GP/BO uncertainty quantification. We utilize the GP framework as a working model, but do not assume correctness of the prior. We instead construct a confidence sequence (CS) for the unknown function using martingale techniques. There is a necessary cost to achieving robustness: if the prior was correct, posterior GP bands are narrower than our CS. Nevertheless, when the prior is wrong, our CS is statistically valid and empirically outperforms standard GP methods, in terms of both coverage and utility for BO. Additionally, we demonstrate that powered likelihoods provide robustness against model misspecification.