Unbiased estimation of log normalizing constants with applications to Bayesian cross-validation

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.