# Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature

arXiv: https://arxiv.org/abs/1411.0439 · HTML: https://mlsanity.com/paper/1411.0439/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Nov 2014.

## Built on by (most cited first, 30 of 31)

- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
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- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Parallel MCMC Without Embarrassing Failures](https://mlsanity.com/paper/2202.11154/) (2022)
- [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)
- [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)
- [PyVBMC: Efficient Bayesian inference in Python](https://mlsanity.com/paper/2303.09519/) (2023)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Fast and robust Bayesian Inference using Gaussian Processes with GPry](https://mlsanity.com/paper/2211.02045/) (2022)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [Marginalising over Stationary Kernels with Bayesian Quadrature](https://mlsanity.com/paper/2106.07452/) (2021)
- [Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC](https://mlsanity.com/paper/2104.03942/) (2021)
- [Fast post-process Bayesian inference with Variational Sparse Bayesian Quadrature](https://mlsanity.com/paper/2303.05263/) (2023)
- [Bayesian Numerical Integration with Neural Networks](https://mlsanity.com/paper/2305.13248/) (2023)
- [Gaussian Process Sampling and Optimization with Approximate Upper and Lower Bounds](https://mlsanity.com/paper/2110.12087/) (2021)
- [Invariant Priors for Bayesian Quadrature](https://mlsanity.com/paper/2112.01578/) (2021)
- [ProEval: Proactive Failure Discovery and Efficient Performance Estimation for Generative AI Evaluation](https://mlsanity.com/paper/2604.23099/) (2026)
- [Bayesian Quadrature for Neural Ensemble Search](https://mlsanity.com/paper/2303.08874/) (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)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Parallelized Acquisition for Active Learning using Monte Carlo Sampling](https://mlsanity.com/paper/2305.19267/) (2023)
- [Respecting the limit:Bayesian optimization with a bound on the optimal value](https://mlsanity.com/paper/2411.04744/) (2024)
- [Measuring the right thing: justifying metrics in AI impact assessments](https://mlsanity.com/paper/2504.05007/) (2025)
- [Sampling as Bandits: Evaluation-Efficient Design for Black-Box Densities](https://mlsanity.com/paper/2509.01437/) (2025)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation](https://mlsanity.com/paper/2606.02247/) (2026)

## Abstract (arXiv)

We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo (MCMC) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function). MCMC has provided approaches to numerical integration that deliver state-of-the-art inference, but can suffer from sample inefficiency and poor convergence diagnostics. Bayesian quadrature techniques offer a model-based solution to such problems, but their uptake has been hindered by prohibitive computation costs. We introduce a warped model for probabilistic integrands (likelihoods) that are known to be non-negative, permitting a cheap active learning scheme to optimally select sample locations. Our algorithm is demonstrated to offer faster convergence (in seconds) relative to simple Monte Carlo and annealed importance sampling on both synthetic and real-world examples.