# Adaptive quadrature schemes for Bayesian inference via active learning

arXiv: https://arxiv.org/abs/2006.00535 · HTML: https://mlsanity.com/paper/2006.00535/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2020.

## Built on by (most cited first, 6 of 6)

- [Bayesian Active Learning with Fully Bayesian Gaussian Processes](https://mlsanity.com/paper/2205.10186/) (2022)
- [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)
- [Fast and robust Bayesian Inference using Gaussian Processes with GPry](https://mlsanity.com/paper/2211.02045/) (2022)
- [Optimality in Noisy Importance Sampling](https://mlsanity.com/paper/2201.02432/) (2022)
- [Robust Bayesian Optimization via Localized Online Conformal Prediction](https://mlsanity.com/paper/2411.17387/) (2024)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)

## Abstract (arXiv)

Numerical integration and emulation are fundamental topics across scientific fields. We propose novel adaptive quadrature schemes based on an active learning procedure. We consider an interpolative approach for building a surrogate posterior density, combining it with Monte Carlo sampling methods and other quadrature rules. The nodes of the quadrature are sequentially chosen by maximizing a suitable acquisition function, which takes into account the current approximation of the posterior and the positions of the nodes. This maximization does not require additional evaluations of the true posterior. We introduce two specific schemes based on Gaussian and Nearest Neighbors (NN) bases. For the Gaussian case, we also provide a novel procedure for fitting the bandwidth parameter, in order to build a suitable emulator of a density function. With both techniques, we always obtain a positive estimation of the marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance sampling interpretation is also described, which allows the design of extended schemes. Several theoretical results are provided and discussed. Numerical results show the advantage of the proposed approach, including a challenging inference problem in an astronomic dynamical model, with the goal of revealing the number of planets orbiting a star.