The marginal likelihood, or Bayesian evidence, is a crucial quantity for Bayesian model comparison but its computation can be challenging for complex models, even in parameters space of moderate dimension. The learned harmonic mean estimator has been shown to provide accurate and robust estimates of the marginal likelihood simply using posterior samples. It is agnostic to the sampling strategy, meaning that the samples can be obtained using any method. This enables marginal likelihood calculation and model comparison with whatever sampling is most suitable for the task. However, the internal density estimators considered previously for the learned harmonic mean can struggle with highly multimodal posteriors. In this work we introduce flow matching-based continuous normalizing flows as a powerful architecture for the internal density estimation of the learned harmonic mean. We demonstrate the ability to handle challenging multimodal posteriors, including an example in 20 parameter dimensions, showcasing the method's ability to handle complex posteriors without the need for fine-tuning or heuristic modifications to the base distribution.

翻译：边缘似然，或称贝叶斯证据，是贝叶斯模型比较的关键量，但其计算对于复杂模型而言具有挑战性，即使在参数空间维度适中的情况下亦如此。学习调和平均估计器已被证明仅通过后验样本即可提供准确稳健的边缘似然估计。它对采样策略具有不可知性，这意味着样本可以通过任何方法获得。这使得边缘似然计算和模型比较能够采用最适合任务的任意采样方式。然而，先前为学习调和平均考虑的内部密度估计器在处理高度多模态的后验分布时可能面临困难。在本工作中，我们引入基于流匹配的连续归一化流作为学习调和平均内部密度估计的强大架构。我们展示了该方法处理具有挑战性的多模态后验分布的能力，包括一个20维参数空间的示例，彰显了该方法无需对基分布进行精细调整或启发式修改即可处理复杂后验分布的优势。