Bayesian model comparison for simulation-based inference

TL;DR

This paper introduces a flexible method for computing Bayesian model evidence in simulation-based inference using the learnt harmonic mean estimator, applicable across various neural density estimation techniques and validated on multiple problems.

Contribution

It develops a novel, sampler-agnostic approach for Bayesian model comparison in likelihood-free inference using the learnt harmonic mean estimator.

Findings

Effective in likelihood-free and likelihood-based settings

Validated on gravitational wave and other inference problems

Compatible with neural posterior, likelihood, and ratio estimation

Abstract

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.