# Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination

arXiv: https://arxiv.org/abs/2206.04734 · HTML: https://mlsanity.com/paper/2206.04734/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 103 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 30 of 32)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Maximizing acquisition functions for Bayesian optimization](https://mlsanity.com/paper/1805.10196/) (2018)
- [Bayesian Batch Active Learning as Sparse Subset Approximation](https://mlsanity.com/paper/1908.02144/) (2019)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Automated Scalable Bayesian Inference via Hilbert Coresets](https://mlsanity.com/paper/1710.05053/) (2017)
- [Batched Gaussian Process Bandit Optimization via Determinantal Point Processes](https://mlsanity.com/paper/1611.04088/) (2016)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [A Multi-Batch L-BFGS Method for Machine Learning](https://mlsanity.com/paper/1605.06049/) (2016)
- [Sparse Variational Inference: Bayesian Coresets from Scratch](https://mlsanity.com/paper/1906.03329/) (2019)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
- [Variational Bayesian Monte Carlo with Noisy Likelihoods](https://mlsanity.com/paper/2006.08655/) (2020)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [Variational Bayesian Monte Carlo](https://mlsanity.com/paper/1810.05558/) (2018)
- [Dynamic nested sampling: an improved algorithm for parameter estimation and evidence calculation](https://mlsanity.com/paper/1704.03459/) (2017)
- [Improving Quadrature for Constrained Integrands](https://mlsanity.com/paper/1802.04782/) (2018)
- [Predicting battery end of life from solar off-grid system field data using machine learning](https://mlsanity.com/paper/2107.13856/) (2021)
- [Batch Selection for Parallelisation of Bayesian Quadrature](https://mlsanity.com/paper/1812.01553/) (2018)
- [Bayesian Parameter Estimation Applied to the Li-ion Battery Single Particle Model with Electrolyte Dynamics](https://mlsanity.com/paper/2001.09890/) (2020)
- [A Randomized Algorithm to Reduce the Support of Discrete Measures](https://mlsanity.com/paper/2006.01757/) (2020)
- [Identifiability and parameter estimation of the single particle lithium-ion battery model](https://mlsanity.com/paper/1702.02471/) (2017)
- [Automated Model Selection with Bayesian Quadrature](https://mlsanity.com/paper/1902.09724/) (2019)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Collaborative Nested Sampling: Big Data vs. complex physical models](https://mlsanity.com/paper/1707.04476/) (2017)

## Built on by (most cited first, 20 of 20)

- [Looping in the Human Collaborative and Explainable Bayesian Optimization](https://mlsanity.com/paper/2310.17273/) (2023)
- [Sampling-based Nystr"om Approximation and Kernel Quadrature](https://mlsanity.com/paper/2301.09517/) (2023)
- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [Bayesian Model Selection of Lithium-Ion Battery Models via Bayesian Quadrature](https://mlsanity.com/paper/2210.17299/) (2022)
- [SOBER: Highly Parallel Bayesian Optimization and Bayesian Quadrature over Discrete and Mixed Spaces](https://mlsanity.com/paper/2301.11832/) (2023)
- [Machine Learning Benchmarks for the Classification of Equivalent Circuit Models from Electrochemical Impedance Spectra](https://mlsanity.com/paper/2302.03362/) (2023)
- [Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach](https://mlsanity.com/paper/2306.05843/) (2023)
- [Principled Bayesian Optimisation in Collaboration with Human Experts](https://mlsanity.com/paper/2410.10452/) (2024)
- [Fast post-process Bayesian inference with Variational Sparse Bayesian Quadrature](https://mlsanity.com/paper/2303.05263/) (2023)
- [A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting](https://mlsanity.com/paper/2404.12219/) (2024)
- [Natural Evolutionary Search meets Probabilistic Numerics](https://mlsanity.com/paper/2507.07288/) (2025)
- [BioBO: Biology-informed Bayesian Optimization for Perturbation Design](https://mlsanity.com/paper/2509.19988/) (2025)
- [Bayesian Quadrature for Neural Ensemble Search](https://mlsanity.com/paper/2303.08874/) (2023)
- [Compressed online Sinkhorn](https://mlsanity.com/paper/2310.05019/) (2023)
- [Policy Gradient with Kernel Quadrature](https://mlsanity.com/paper/2310.14768/) (2023)
- [Normalizing Flow Regression for Bayesian Inference with Offline Likelihood Evaluations](https://mlsanity.com/paper/2504.11554/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Measuring the right thing: justifying metrics in AI impact assessments](https://mlsanity.com/paper/2504.05007/) (2025)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [Hierarchical Bayesian Quadrature](https://mlsanity.com/paper/2607.10793/) (2026)

## Abstract (arXiv)

Calculation of Bayesian posteriors and model evidences typically requires numerical integration. Bayesian quadrature (BQ), a surrogate-model-based approach to numerical integration, is capable of superb sample efficiency, but its lack of parallelisation has hindered its practical applications. In this work, we propose a parallelised (batch) BQ method, employing techniques from kernel quadrature, that possesses an empirically exponential convergence rate. Additionally, just as with Nested Sampling, our method permits simultaneous inference of both posteriors and model evidence. Samples from our BQ surrogate model are re-selected to give a sparse set of samples, via a kernel recombination algorithm, requiring negligible additional time to increase the batch size. Empirically, we find that our approach significantly outperforms the sampling efficiency of both state-of-the-art BQ techniques and Nested Sampling in various real-world datasets, including lithium-ion battery analytics.