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。这类积分对于证据估计至关重要，而证据估计在模型选择、数值分析等多个领域具有重要意义。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法（Llorente等，2020）。本文证明，Yakowitz（1978）提出的黎曼和估计量可在嵌套采样（Skilling，2006）框架下实现 $O(n^{-4})$ 收敛速度，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、嵌套采样算法及其与垂直似然蒙特卡洛方法关联的文献。通过理论与数值论证，展示了这两种思想的融合如何在更高维空间中产生更优且更稳健的证据估计量。同时，我们还讨论了模拟洛伦兹曲线的思路，该曲线可规避对垂直表示法和嵌套采样至关重要的 $\Lambda$ 函数难以计算的问题。