  Posterior distributions often feature intractable normalizing constants,
called marginal likelihoods or evidence, that are useful for model comparison
via Bayes factors. This has motivated a number of methods for estimating ratios
of normalizing constants in statistics. In computational physics the logarithm
of these ratios correspond to free energy differences. Combining unbiased
Markov chain Monte Carlo estimators with path sampling, also called
thermodynamic integration, we propose new unbiased estimators of the logarithm
of ratios of normalizing constants. As a by-product, we propose unbiased
estimators of the Bayesian cross-validation criterion. The proposed estimators
are consistent, asymptotically Normal and can easily benefit from parallel
processing devices. Various examples are considered for illustration.
