I'm trying to compute the marginal likelihood for a statistical model by Monte Carlo methods:
$$f(x) = \int f(x\mid\theta) \pi(\theta)\, d\theta$$
The likelihood is well behaved - smooth, log-concave - but high-dimensional. I've tried importance sampling, but the results are wonky and depend highly on the proposal I'm using. I briefly considered doing Hamiltonian Monte Carlo to compute posterior samples assuming a uniform prior over $\theta$ and taking the harmonic mean, until I saw this. Lesson learned, the harmonic mean can have infinite variance. Is there an alternative MCMC estimator that is nearly as simple, but has a well-behaved variance?