Efficient Bayesian model selection relies on the model evidence or marginal likelihood, whose computation often requires evaluating an intractable integral. The harmonic mean estimator (HME) has long been a standard method of approximating the evidence. While computationally simple, the version introduced by Newton and Raftery (1994) potentially suffers from infinite variance. To overcome this issue,Gelfand and Dey (1994) defined a standardized representation of the estimator based on an instrumental function and Robert and Wraith (2009) later proposed to use higher posterior density (HPD) indicators as instrumental functions. Following this approach, a practical method is proposed, based on an elliptical covering of the HPD region with non-overlapping ellipsoids. The resulting estimator, called the Elliptical Covering Marginal Likelihood Estimator (ECMLE), not only eliminates the infinite-variance issue of the original HME and allows exact volume computations, but is also able to be used in multimodal settings. Through several examples, we illustrate that ECMLE outperforms other recent methods such as THAMES and its improved version (Metodiev et al 2024, 2025). Moreover, ECMLE demonstrates lower variance, a key challenge that subsequent HME variants have sought to address, and provides more stable evidence approximations, even in challenging settings.

翻译：高效的贝叶斯模型选择依赖于模型证据（即边际似然），其计算通常需要求解难以处理的积分。调和平均估计量（HME）长期以来是近似证据的标准方法。虽然计算简单，但Newton与Raftery（1994）提出的原始版本可能存在无限方差问题。为克服此缺陷，Gelfand与Dey（1994）基于工具函数定义了该估计量的标准化表示，随后Robert与Wraith（2009）提出采用高后验密度（HPD）指示函数作为工具函数。沿此思路，本文提出一种基于HPD区域椭圆覆盖的实用方法——通过非重叠椭球体对HPD区域进行椭圆覆盖。由此得到的估计量称为椭圆覆盖边际似然估计量（ECMLE），它不仅消除了原始HME的无限方差问题、支持精确体积计算，还能适用于多模态场景。通过多个案例研究表明，ECMLE在性能上优于THAMES等近期方法及其改进版本（Metodiev等，2024，2025）。此外，ECMLE展现出更低的方差（这是后续HME变体致力解决的关键挑战），即使在复杂场景中也能提供更稳定的证据近似结果。