Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

LP-based techniques struggling to scale favorably in large instances.

These methods can be broadly categorized into two types: function learning and operator learning approaches. In function learning, the goal is to directly learn the solution.

As such, in games where the marginalization of CCE coincides with the game's Nash equilibria (like

A common feature of these applications is that there are multiple decision makers interacting with each other in a common environment.

We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations.

We demonstrate that these limitations can be sidestepped by reframing the generative process as a discrete optimal control episode.