# Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization

arXiv: https://arxiv.org/abs/2605.20145 · HTML: https://mlsanity.com/paper/2605.20145/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 16 of 16)

- [Misspecified Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2111.05008/) (2021)
- [Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination](https://mlsanity.com/paper/2108.03210/) (2021)
- [Bayesian Optimization with Conformal Prediction Sets](https://mlsanity.com/paper/2210.12496/) (2022)
- [A Bayesian approach to constrained single- and multi-objective optimization](https://mlsanity.com/paper/1510.00503/) (2015)
- [Bayesian Quantile and Expectile Optimisation](https://mlsanity.com/paper/2001.04833/) (2020)
- [Confidence on the Focal: Conformal Prediction with Selection-Conditional Coverage](https://mlsanity.com/paper/2403.03868/) (2024)
- [Uncertainty quantification using martingales for misspecified Gaussian processes](https://mlsanity.com/paper/2006.07368/) (2020)
- [Online Calibrated and Conformal Prediction Improves Bayesian Optimization](https://mlsanity.com/paper/2112.04620/) (2021)
- [Ordinal Bayesian Optimisation](https://mlsanity.com/paper/1912.02493/) (2019)
- [Uncertainty Quantification for Bayesian Optimization](https://mlsanity.com/paper/2002.01569/) (2020)
- [Tail calibration of probabilistic forecasts](https://mlsanity.com/paper/2407.03167/) (2024)
- [Ranking over Regression for Bayesian Optimization and Molecule Selection](https://mlsanity.com/paper/2410.09290/) (2024)
- [Relaxed Gaussian process interpolation: a goal-oriented approach to Bayesian optimization](https://mlsanity.com/paper/2206.03034/) (2022)
- [CRPS-Based Targeted Sequential Design with Application in Chemical Space](https://mlsanity.com/paper/2503.11250/) (2025)
- [Design-marginal calibration of Gaussian process predictive distributions: Bayesian and conformal approaches](https://mlsanity.com/paper/2512.05611/) (2025)
- [Robust Bayesian Optimization via Localized Online Conformal Prediction](https://mlsanity.com/paper/2411.17387/) (2024)

## Abstract (arXiv)

Gaussian process (GP) predictive distributions are commonly used in Bayesian optimization (BO) to guide the selection of evaluation points for expensive objective functions. The choice of kernel and hyperparameters has a strong influence on the exploration--exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on both the probability mass below the current best value and the shape of the predictive distribution in this region. This article studies goal-oriented calibration of GP predictive distributions below a low threshold t in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. We consider two complementary forms of calibration below t for inputs distributed according to a reference measure μ: occurrence calibration over the design space and thresholded μ-calibration on sublevel sets of the form x∈mathbbX, f(x)≤ t. We propose tcGP, a post-hoc method that combines these two forms of calibration for GP predictive distributions below t. With fixed GP hyperparameters, the exact EI sampling criterion based on tcGP generates a sequence of evaluation points that is dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.