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