Adaptive quadrature schemes for Bayesian inference via active learning

TL;DR

This paper introduces adaptive quadrature schemes for Bayesian inference that utilize active learning to efficiently select evaluation points, improving posterior approximation and evidence estimation without extra true posterior evaluations.

Contribution

It presents novel active learning-based adaptive quadrature methods using Gaussian and Nearest Neighbors bases, with a new bandwidth fitting procedure and theoretical analysis.

Findings

Methods provide positive estimates of Bayesian evidence.

Numerical results outperform traditional methods in complex inference tasks.

Approach is effective in challenging astronomical inference problems.

Abstract

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.