Model Evidence with Fast Tree Based Quadrature

TL;DR

The paper introduces Tree Quadrature (TQ), a flexible high-dimensional integration algorithm that constructs surrogate models with regression trees, enabling the use of advanced sampling methods and outperforming traditional techniques in higher dimensions.

Contribution

TQ separates sampling from integration approximation, allowing the use of any sampling method and combining it with existing integrators for improved accuracy in high dimensions.

Findings

TQ achieves accurate high-dimensional integrals up to 15 dimensions.

TQ outperforms Monte Carlo and Vegas in dimensions 4 and above.

TQ effectively integrates with various sampling algorithms.

Abstract

High dimensional integration is essential to many areas of science, ranging from particle physics to Bayesian inference. Approximating these integrals is hard, due in part to the difficulty of locating and sampling from regions of the integration domain that make significant contributions to the overall integral. Here, we present a new algorithm called Tree Quadrature (TQ) that separates this sampling problem from the problem of using those samples to produce an approximation of the integral. TQ places no qualifications on how the samples provided to it are obtained, allowing it to use state-of-the-art sampling algorithms that are largely ignored by existing integration algorithms. Given a set of samples, TQ constructs a surrogate model of the integrand in the form of a regression tree, with a structure optimised to maximise integral precision. The tree divides the integration domain into smaller containers, which are individually integrated and aggregated to estimate the overall integral. Any method can be used to integrate each individual container, so existing integration methods, like Bayesian Monte Carlo, can be combined with TQ to boost their performance. On a set of benchmark problems, we show that TQ provides accurate approximations to integrals in up to 15 dimensions; and in dimensions 4 and above, it outperforms simple Monte Carlo and the popular Vegas method.