  Bayesian model comparison relies upon the model evidence, yet for many models
of interest the model evidence is unavailable in closed form and must be
approximated. Many of the estimators for evidence that have been proposed in
the Monte Carlo literature suffer from high variability. This paper considers
the reduction of variance that can be achieved by exploiting control variates
in this setting. Our methodology is based on thermodynamic integration and
applies whenever the gradient of both the log-likelihood and the log-prior with
respect to the parameters can be efficiently evaluated. Results obtained on
regression models and popular benchmark datasets demonstrate a significant and
sometimes dramatic reduction in estimator variance and provide insight into the
wider applicability of control variates to Bayesian model comparison.

Methodology