# Bayesian Quadrature for Multiple Related Integrals

arXiv: https://arxiv.org/abs/1801.04153 · HTML: https://mlsanity.com/paper/1801.04153/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2018.

## Built on by (most cited first, 16 of 16)

- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Marginalising over Stationary Kernels with Bayesian Quadrature](https://mlsanity.com/paper/2106.07452/) (2021)
- [Multi-fidelity Monte Carlo: a pseudo-marginal approach](https://mlsanity.com/paper/2210.01534/) (2022)
- [Bayesian Numerical Integration with Neural Networks](https://mlsanity.com/paper/2305.13248/) (2023)
- [Invariant Priors for Bayesian Quadrature](https://mlsanity.com/paper/2112.01578/) (2021)
- [Practical Bayesian Algorithm Execution via Posterior Sampling](https://mlsanity.com/paper/2410.20596/) (2024)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Integrated Gaussian Processes for Robust and Adaptive Multi-Object Tracking](https://mlsanity.com/paper/2507.04116/) (2025)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Amortized quadrature for posterior expectations in inverse problems](https://mlsanity.com/paper/2606.15871/) (2026)
- [Amortised adaptive importance sampling](https://mlsanity.com/paper/2610.16736/) (2026)

## Abstract (arXiv)

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.