  We propose an easily computed estimator of marginal likelihoods from
posterior simulation output, via reciprocal importance sampling, combining
earlier proposals of DiCiccio et al (1997) and Robert and Wraith (2009). This
involves only the unnormalized posterior densities from the sampled parameter
values, and does not involve additional simulations beyond the main posterior
simulation, or additional complicated calculations. It is unbiased for the
reciprocal of the marginal likelihood, consistent, has finite variance, and is
asymptotically normal. It involves one user-specified control parameter, and we
derive an optimal way of specifying this. We illustrate it with several
numerical examples.
