Research

I work on uncertainty quantification for expensive experiments and numerical simulations. My principal interests are function approximation using Gaussian processes, sequential methods for choosing new simulations efficiently, Bayesian optimization, reliability analysis, set inversion, and predictive calibration.

Function Approximation

Gaussian processes provide probabilistic models for approximating an unknown function from a limited number of evaluations. I study the choice of covariance functions and parameters, regularized regression, numerical robustness, and diagnostic methods. This work also includes conditional simulation and the assessment of predictive distributions.

Current work includes calibration and multi-fidelity experiments. It also addresses high-dimensional problems and problems involving large numbers of observations. The aim is to assess pointwise accuracy and to determine whether the predictive distribution is appropriate for the quantity to be estimated or the decision to be made.

Sequential Design

In sequential design, the observations already obtained are used to choose the next simulation. The choice can be formulated as a Bayesian decision problem in which the expected uncertainty remaining about a specified quantity or decision is minimized.

Stepwise Uncertainty Reduction (SUR) has been applied to excursion-set estimation, failure-probability estimation, reliable set inversion, and quantile-set inversion. The articles on failure-probability estimation (2012) and parallel SUR for excursion sets (2014) develop sequential and batch-sequential strategies. Bayesian subset simulation (2017) combines sequential design based on Gaussian processes with subset simulation to estimate small failure probabilities.

The same decision principle applies to multi-fidelity computer experiments, where the information expected from a simulation is weighed against its cost.

Bayesian Optimization

Work on the Bayesian optimization of expensive-to-evaluate functions includes an informational criterion for global optimization (2009), convergence properties of expected improvement (2010), a fully Bayesian expected improvement criterion (2011), and constrained single- and multi-objective optimization (2017). Recent results establish simple-regret rates and minimax optimality for expected improvement with a fixed prior. These results apply to functions in the reproducing kernel Hilbert spaces associated with Matérn and squared-exponential kernels.

Recent work examines the calibration of the lower tail of the Gaussian-process predictive distribution used to compute expected improvement. In minimization, this tail directly governs the balance between exploration and exploitation. Its calibration must therefore be considered in relation to the optimization objective.

Predictive Calibration

A Gaussian-process predictive distribution need not be calibrated, particularly when covariance parameters are selected from the data. I study Bayesian and conformal methods for improving coverage and probabilistic calibration while retaining predictive distributions that can be used in sequential algorithms.

I am studying design-marginal calibration, goal-oriented calibration, and cross-validation diagnostics. These studies examine the relation between the statistical assessment of predictive distributions and their use in decision-making.

Scientific Software

I also develop open-source software. GPmp provides routines for function approximation using Gaussian processes, including interpolation and regression, parameter selection, diagnostics, and fast leave-one-out cross-validation. It also provides routines for conditional simulation and posterior exploration.

The gpmp-contrib package implements higher-level methods for Bayesian optimization, excursion-set estimation, and set inversion. I previously contributed to STK, a MATLAB/Octave toolbox for kriging.

PhD students

PhD students supervised or co-supervised, in chronological order:

• Miguel Piera-Martinez
• Julien Villemonteix
• Aimad El Habachi
• Sándor Bilicz
• Ling Li
• Romain Benassi
• Paul Féliot
• Rémi Stroh
• Bruno Tebbal Barracosa
• Sébastien Petit
• Subhasish Basak
• Romain Ait Abdelmalek-Lomenech
• Boris Kratz
• Enikő Bartók
• Aurélien Pion
• Mohamed Badi

Selected publications

• 2026 — A. Pion and E. Vazquez, Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization, ICML 2026.
• 2026 — E. Vazquez and S. Petit, Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs, preprint.
• 2025 — A. Pion and E. Vazquez, Design-Marginal Calibration of Gaussian Process Predictive Distributions: Bayesian and Conformal Approaches, preprint.
• 2025 — S. J. Petit, J. Bect and E. Vazquez, Relaxed Gaussian Process Interpolation: a Goal-Oriented Approach to Bayesian Optimization, Journal of Machine Learning Research, 26(195), 1–70.
• 2025 — R. Ait Abdelmalek-Lomenech, J. Bect, V. Chabridon and E. Vazquez, Bayesian Sequential Design of Computer Experiments for Quantile Set Inversion, Technometrics, 67(1), 112–121.
• 2023 — S. J. Petit, J. Bect, P. Feliot and E. Vazquez, Parameter Selection in Gaussian Process Interpolation: An Empirical Study of Selection Criteria, SIAM/ASA Journal on Uncertainty Quantification, 11(4), 1308–1328.
• 2022 — R. Stroh, J. Bect, S. Demeyer, N. Fischer, D. Marquis and E. Vazquez, Sequential Design of Multi-Fidelity Computer Experiments: Maximizing the Rate of Stepwise Uncertainty Reduction, Technometrics, 64(2), 199–209.
• 2017 — P. Feliot, J. Bect and E. Vazquez, A Bayesian Approach to Constrained Single- and Multi-Objective Optimization, Journal of Global Optimization, 67(1), 97–133.
• 2017 — J. Bect, L. Li and E. Vazquez, Bayesian Subset Simulation, SIAM/ASA Journal on Uncertainty Quantification, 5(1), 762–786.
• 2014 — C. Chevalier, J. Bect, D. Ginsbourger, E. Vazquez, V. Picheny and Y. Richet, Fast Parallel Kriging-Based Stepwise Uncertainty Reduction With Application to the Identification of an Excursion Set, Technometrics, 56(4), 455–465.
• 2012 — J. Bect, D. Ginsbourger, L. Li, V. Picheny and E. Vazquez, Sequential Design of Computer Experiments for the Estimation of a Probability of Failure, Statistics and Computing, 22, 773–793.
• 2011 — R. Benassi, J. Bect and E. Vazquez, Robust Gaussian process-based global optimization using a fully Bayesian expected improvement criterion, Learning and Intelligent Optimization, Lecture Notes in Computer Science 6683, 176–190.
• 2010 — E. Vazquez and J. Bect, Convergence properties of the expected improvement algorithm with fixed mean and covariance functions, Journal of Statistical Planning and Inference, 140(11), 3088–3095.
• 2009 — J. Villemonteix, E. Vazquez and E. Walter, An Informational Approach to the Global Optimization of Expensive-to-Evaluate Functions, Journal of Global Optimization, 44(4), 509–534.

See the Publications tab for the complete list.

Last updated: October 2026.