Learned harmonic mean estimation of the marginal likelihood for multimodal posteriors with flow matching

TL;DR

This paper introduces a flow matching-based approach to improve the learned harmonic mean estimator for marginal likelihood, enabling accurate Bayesian model comparison even with complex, multimodal posteriors in high-dimensional spaces.

Contribution

It proposes a novel flow matching architecture for internal density estimation in the learned harmonic mean method, enhancing performance on challenging multimodal posteriors.

Findings

Effective handling of highly multimodal posteriors

Successful application in 20-dimensional parameter space

Robust and accurate marginal likelihood estimates

Abstract

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.