# Improving Quadrature for Constrained Integrands

arXiv: https://arxiv.org/abs/1802.04782 · HTML: https://mlsanity.com/paper/1802.04782/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2018.

## Built on by (most cited first, 12 of 12)

- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [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)
- [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)
- [Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC](https://mlsanity.com/paper/2104.03942/) (2021)
- [Invariant Priors for Bayesian Quadrature](https://mlsanity.com/paper/2112.01578/) (2021)
- [Bayesian Quadrature for Neural Ensemble Search](https://mlsanity.com/paper/2303.08874/) (2023)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Scalable Bayesian Transformed Gaussian Processes](https://mlsanity.com/paper/2210.10973/) (2022)
- [Parallelized Acquisition for Active Learning using Monte Carlo Sampling](https://mlsanity.com/paper/2305.19267/) (2023)

## Abstract (arXiv)

We present an improved Bayesian framework for performing inference of affine transformations of constrained functions. We focus on quadrature with nonnegative functions, a common task in Bayesian inference. We consider constraints on the range of the function of interest, such as nonnegativity or boundedness. Although our framework is general, we derive explicit approximation schemes for these constraints, and argue for the use of a log transformation for functions with high dynamic range such as likelihood surfaces. We propose a novel method for optimizing hyperparameters in this framework: we optimize the marginal likelihood in the original space, as opposed to in the transformed space. The result is a model that better explains the actual data. Experiments on synthetic and real-world data demonstrate our framework achieves superior estimates using less wall-clock time than existing Bayesian quadrature procedures.