Consistency of some sequential experimental design strategies for excursion set estimation based on vector-valued Gaussian processes

TL;DR

This paper extends the theoretical framework of sequential experimental design strategies based on Gaussian processes to vector-valued cases, clarifying the connection between continuous processes and Gaussian measures, and analyzing the impact of vector-valued settings on consistency results.

Contribution

It generalizes existing consistency results for Gaussian process-based sequential designs to vector-valued functions, addressing challenges due to lack of continuity in pseudo-inverse mappings.

Findings

Extended consistency results to vector-valued Gaussian processes.

Clarified the connection between continuous Gaussian processes and Gaussian measures in vector settings.

Analyzed the impact of vector-valued settings on properties of sequential design strategies.

Abstract

We tackle the extension to the vector-valued case of consistency results for Stepwise Uncertainty Reduction sequential experimental design strategies established in [Bect et al., A supermartingale approach to Gaussian process based sequential design of experiments, Bernoulli 25, 2019]. This lead us in the first place to clarify, assuming a compact index set, how the connection between continuous Gaussian processes and Gaussian measures on the Banach space of continuous functions carries over to vector-valued settings. From there, a number of concepts and properties from the aforementioned paper can be readily extended. However, vector-valued settings do complicate things for some results, mainly due to the lack of continuity for the pseudo-inverse mapping that affects the conditional mean and covariance function given finitely many pointwise observations. We apply obtained results to the Integrated Bernoulli Variance and the Expected Measure Variance uncertainty functionals employed in [Fossum et al., Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling, The Annals of Applied Statistics 15, 2021] for the estimation for excursion sets of vector-valued functions.