In Bayesian inference, the approximation of integrals of the form $\psi = \mathbb{E}_{F}{l(X)} = \int_{\chi} l(\mathbf{x}) d F(\mathbf{x})$ is a fundamental challenge. Such integrals are crucial for evidence estimation, which is important for various purposes, including model selection and numerical analysis. The existing strategies for evidence estimation are classified into four categories: deterministic approximation, density estimation, importance sampling, and vertical representation (Llorente et al., 2020). In this paper, we show that the Riemann sum estimator due to Yakowitz (1978) can be used in the context of nested sampling (Skilling, 2006) to achieve a $O(n^{-4})$ rate of convergence, faster than the usual Ergodic Central Limit Theorem. We provide a brief overview of the literature on the Riemann sum estimators and the nested sampling algorithm and its connections to vertical likelihood Monte Carlo. We provide theoretical and numerical arguments to show how merging these two ideas may result in improved and more robust estimators for evidence estimation, especially in higher dimensional spaces. We also briefly discuss the idea of simulating the Lorenz curve that avoids the problem of intractable $\Lambda$ functions, essential for the vertical representation and nested sampling.

翻译：在贝叶斯推断中，形如 $\psi = \mathbb{E}_{F}{l(X)} = \int_{\chi} l(\mathbf{x}) d F(\mathbf{x})$ 的积分近似是一项基本挑战。这类积分对证据估计至关重要，而证据估计在模型选择、数值分析等众多领域具有重要价值。现有证据估计策略可分为四类：确定性近似、密度估计、重要性采样和垂直表示（Llorente 等，2020）。本文证明，Yakowitz (1978) 提出的黎曼和估计量可在嵌套抽样（Skilling, 2006）框架下实现 $O(n^{-4})$ 收敛速率，快于常规遍历中心极限定理。我们简要回顾了黎曼和估计量及嵌套抽样算法的相关文献，并阐明其与垂直似然蒙特卡洛方法的关联。通过理论与数值论证，我们展示如何融合这两种思想，从而在证据估计，特别是高维空间中，获得更优且更具鲁棒性的估计量。此外，我们简要讨论了洛伦兹曲线模拟方法，该方法可规避垂直表示与嵌套抽样中不可或缺的 $\Lambda$ 函数难以处理的问题。