Approximating evidence via bounded harmonic means

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.

💡 Research Summary

The paper addresses a long‑standing difficulty in Bayesian model selection: the reliable estimation of the marginal likelihood (model evidence) when the required integral cannot be evaluated analytically. The classic Harmonic Mean Estimator (HME) introduced by Newton and Raftery (1994) is computationally attractive because it only needs posterior samples and likelihood evaluations, but it suffers from potentially infinite variance, especially when the posterior has heavy tails or irregular geometry.

Gelfand and Dey (1994) showed that the HME can be generalized via the identity

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