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 (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 multi-modal settings. Through several examples, we illustrate that ECMLE outperforms other recent methods such as THAMES and Mixture THAMES (Metodiev et al., 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或边缘似然，其计算通常需要评估一个难以处理的积分。调和均值估计器（HME）长期以来一直是近似证据的标准方法。虽然计算简单，但Newton和Raftery（1994）提出的版本可能存在无限方差问题。为克服此问题，Gelfand和Dey（1994）基于一个工具函数定义了该估计器的标准化表示，随后Robert和Wraith（2009）提出使用高后验密度（HPD）指示函数作为工具函数。遵循此思路，本文提出一种基于HPD区域椭圆覆盖的实用方法，该覆盖由互不重叠的椭球构成。所得估计器（ECMLE）不仅消除了原始HME的无限方差问题并允许精确的体积计算，还能用于多模态场景。通过多个示例，我们证明ECMLE优于THAMES和Mixture THAMES（Metodiev等人，2025）等近期方法。此外，ECMLE展现出更低的方差——这是后续HME变体一直试图解决的关键挑战——并在具有挑战性的场景中提供更稳定的证据近似。