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.

中文翻译：

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使用基于快速树的正交模型证据

高维积分对于许多科学领域都是必不可少的，从粒子物理学到贝叶斯推理。逼近这些积分很困难，部分原因是很难从对整体积分做出重大贡献的积分域区域中进行定位和采样。在这里，我们提出了一种称为树正交 (TQ) 的新算法，它将采样问题与使用这些样本生成积分近似的问题分开。TQ 对如何获得提供给它的样本没有任何限制，允许它使用现有集成算法在很大程度上忽略的最先进的采样算法。给定一组样本，TQ 以回归树的形式构建被积函数的代理模型，其结构经过优化以最大化积分精度。树将集成域划分为更小的容器，这些容器被单独集成和聚合以估计整体积分。任何方法都可以用于集成每个单独的容器，因此现有的集成方法，如贝叶斯蒙特卡罗，可以与 TQ 结合以提高其性能。在一组基准问题上，我们表明 TQ 提供了对多达 15 维积分的精确近似值；在维度 4 及以上，它优于简单的 Monte Carlo 和流行的 Vegas 方法。在一组基准问题上，我们表明 TQ 提供了对多达 15 维积分的精确近似值；在维度 4 及以上，它优于简单的 Monte Carlo 和流行的 Vegas 方法。在一组基准问题上，我们表明 TQ 提供了对多达 15 维积分的精确近似值；在维度 4 及以上，它优于简单的 Monte Carlo 和流行的 Vegas 方法。