# Probabilistic Integration: A Role in Statistical Computation?

arXiv: https://arxiv.org/abs/1512.00933 · HTML: https://mlsanity.com/paper/1512.00933/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2015.

## Built on by (most cited first, 26 of 26)

- [Active Surrogate Estimators: An Active Learning Approach to Label-Efficient Model Evaluation](https://mlsanity.com/paper/2202.06881/) (2022)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [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)
- [A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning and ABC](https://mlsanity.com/paper/2108.00490/) (2021)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting](https://mlsanity.com/paper/2404.12219/) (2024)
- [Sparse solutions of the kernel herding algorithm by improved gradient approximation](https://mlsanity.com/paper/2105.07900/) (2021)
- [Keep it Tighter -- A Story on Analytical Mean Embeddings](https://mlsanity.com/paper/2110.09516/) (2021)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Natural Evolutionary Search meets Probabilistic Numerics](https://mlsanity.com/paper/2507.07288/) (2025)
- [Linear-Time Probabilistic Solutions of Boundary Value Problems](https://mlsanity.com/paper/2106.07761/) (2021)
- [Monte Carlo with kernel-based Gibbs measures: Guarantees for probabilistic herding](https://mlsanity.com/paper/2402.11736/) (2024)
- [Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Time](https://mlsanity.com/paper/2404.12923/) (2024)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Fast and Provably Accurate Sequential Designs using Hilbert Space Gaussian Processes](https://mlsanity.com/paper/2604.20414/) (2026)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [Amortized quadrature for posterior expectations in inverse problems](https://mlsanity.com/paper/2606.15871/) (2026)
- [GNet: A scalable and flexible Gaussian process network with nonparametric neurons](https://mlsanity.com/paper/2607.10735/) (2026)
- [Hierarchical Bayesian Quadrature](https://mlsanity.com/paper/2607.10793/) (2026)
- [Counterfactual Evaluation of Temporal Observation Protocols](https://mlsanity.com/paper/2608.22221/) (2026)

## Abstract (arXiv)

A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the presence of an unknown numerical error. Our main technical contribution is to establish, for the first time, rates of posterior contraction for these methods. These show that probabilistic integrators can in principle enjoy the "best of both worlds", leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assess the impact of numerical error on scientific conclusions. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir.