# Conditional Bayesian Quadrature

arXiv: https://arxiv.org/abs/2406.16530 · HTML: https://mlsanity.com/paper/2406.16530/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 97 references that are in the map.
- First posted on arXiv 24 Jun 2024.

## Builds on (most cited first, 30 of 36)

- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On Nesting Monte Carlo Estimators](https://mlsanity.com/paper/1709.06181/) (2017)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Variational Bayesian Monte Carlo](https://mlsanity.com/paper/1810.05558/) (2018)
- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Sampling-based Nystr"om Approximation and Kernel Quadrature](https://mlsanity.com/paper/2301.09517/) (2023)
- [Kernel Mean Shrinkage Estimators](https://mlsanity.com/paper/1405.5505/) (2014)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [Vector-Valued Control Variates](https://mlsanity.com/paper/2109.08944/) (2021)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [When is Importance Weighting Correction Needed for Covariate Shift Adaptation?](https://mlsanity.com/paper/2303.04020/) (2023)
- [Discrepancy-based Inference for Intractable Generative Models using Quasi-Monte Carlo](https://mlsanity.com/paper/2106.11561/) (2021)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Active Uncertainty Calibration in Bayesian ODE Solvers](https://mlsanity.com/paper/1605.03364/) (2016)
- [Symmetry Exploits for Bayesian Cubature Methods](https://mlsanity.com/paper/1809.10227/) (2018)
- [Multilevel Bayesian Quadrature](https://mlsanity.com/paper/2210.08329/) (2022)
- [Characteristic Kernels and Infinitely Divisible Distributions](https://mlsanity.com/paper/1403.7304/) (2014)
- [Detecting and diagnosing prior and likelihood sensitivity with power-scaling](https://mlsanity.com/paper/2107.14054/) (2021)

## Built on by (most cited first, 9 of 9)

- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [ActiveCQ: Active Estimation of Causal Quantities](https://mlsanity.com/paper/2509.24293/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Amortised adaptive importance sampling](https://mlsanity.com/paper/2610.16736/) (2026)

## Abstract (arXiv)

We propose a novel approach for estimating conditional or parametric expectations in the setting where obtaining samples or evaluating integrands is costly. Through the framework of probabilistic numerical methods (such as Bayesian quadrature), our novel approach allows to incorporates prior information about the integrands especially the prior smoothness knowledge about the integrands and the conditional expectation. As a result, our approach provides a way of quantifying uncertainty and leads to a fast convergence rate, which is confirmed both theoretically and empirically on challenging tasks in Bayesian sensitivity analysis, computational finance and decision making under uncertainty.