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Computing the marginal likelihood (also called the Bayesian model evidence)is an important task in Bayesian model selection, providing a principledquantitative way to compare models. The learned harmonic mean estimator solvesthe exploding variance problem of the original harmonic mean estimation of themarginal likelihood. The learned harmonic mean estimator learns an importancesampling target distribution that approximates the optimal distribution. Whilethe approximation need not be highly accurate, it is critical that theprobability mass of the learned distribution is contained within the posteriorin order to avoid the exploding variance problem. In previous work a bespokeoptimization problem is introduced when training models in order to ensure thisproperty is satisfied. In the current article we introduce the use ofnormalizing flows to represent the importance sampling target distribution. Aflow-based model is trained on samples from the posterior by maximum likelihoodestimation. Then, the probability density of the flow is concentrated bylowering the variance of the base distribution, i.e. by lowering its"temperature", ensuring its probability mass is contained within the posterior.This approach avoids the need for a bespoke optimisation problem and carefulfine tuning of parameters, resulting in a more robust method. Moreover, the useof normalizing flows has the potential to scale to high dimensional settings.We present preliminary experiments demonstrating the effectiveness of the useof flows for the learned harmonic mean estimator. The harmonic codeimplementing the learned harmonic mean, which is publicly available, has beenupdated to now support normalizing flows.

Learned harmonic mean estimation of the marginal likelihood with normalizing flows