Machine learning assisted Bayesian model comparison: learnt harmonic\n mean estimator

We resurrect the infamous harmonic mean estimator for computing the marginal\nlikelihood (Bayesian evidence) and solve its problematic large variance. The\nmarginal likelihood is a key component of Bayesian model selection to evaluate\nmodel posterior probabilities; however, its computation is challenging. The\noriginal harmonic mean estimator, first proposed by Newton and Raftery in 1994,\ninvolves computing the harmonic mean of the likelihood given samples from the\nposterior. It was immediately realised that the original estimator can fail\ncatastrophically since its variance can become very large (possibly not\nfinite). A number of variants of the harmonic mean estimator have been proposed\nto address this issue although none have proven fully satisfactory. We present\nthe \\emph{learnt harmonic mean estimator}, a variant of the original estimator\nthat solves its large variance problem. This is achieved by interpreting the\nharmonic mean estimator as importance sampling and introducing a new target\ndistribution. The new target distribution is learned to approximate the optimal\nbut inaccessible target, while minimising the variance of the resulting\nestimator. Since the estimator requires samples of the posterior only, it is\nagnostic to the sampling strategy used. We validate the estimator on a variety\nof numerical experiments, including a number of pathological examples where the\noriginal harmonic mean estimator fails catastrophically. We also consider a\ncosmological application, where our approach leads to $\\sim$ 3 to 6 times more\nsamples than current state-of-the-art techniques in 1/3 of the time. In all\ncases our learnt harmonic mean estimator is shown to be highly accurate. The\nestimator is computationally scalable and can be applied to problems of\ndimension $O(10^3)$ and beyond. Code implementing the learnt harmonic mean\nestimator is made publicly available\n