  Comparison of appropriate models to describe observational data is a
fundamental task of science. The Bayesian model evidence, or marginal
likelihood, is a computationally challenging, yet crucial, quantity to estimate
to perform Bayesian model comparison. We introduce a methodology to compute the
Bayesian model evidence in simulation-based inference (SBI) scenarios (also
often called likelihood-free inference). In particular, we leverage the
recently proposed learnt harmonic mean estimator and exploit the fact that it
is decoupled from the method used to generate posterior samples, i.e. it
requires posterior samples only, which may be generated by any approach. This
flexibility, which is lacking in many alternative methods for computing the
model evidence, allows us to develop SBI model comparison techniques for the
three main neural density estimation approaches, including neural posterior
estimation (NPE), neural likelihood estimation (NLE), and neural ratio
estimation (NRE). We demonstrate and validate our SBI evidence calculation
techniques on a range of inference problems, including a gravitational wave
example. Moreover, we further validate the accuracy of the learnt harmonic mean
estimator, implemented in the HARMONIC software, in likelihood-based settings.
These results highlight the potential of HARMONIC as a sampler-agnostic method
to estimate the model evidence in both likelihood-based and simulation-based
scenarios.
