On Monte Carlo methods for estimating ratios of normalizing constants

Recently, estimating ratios of normalizing constants has played an important role in Bayesian computations. Applications of estimating ratios of normalizing constants arise in many aspects of Bayesian statistical inference. In this article, we present an overview and discuss the current Monte Carlo methods for estimating ratios of normalizing constants. Then we propose a new ratio importance sampling method and establish its theoretical framework. We find that the ratio importance sampling method can be better than the current methods, for example, the bridge sampling method (Meng and Wong) and the path sampling method (Gelman and Meng), in the sense of minimizing asymptotic relative mean-square errors of estimators. An example is given for illustrative purposes. Finally, we present two special applications and the general implementation issues for estimating ratios of normalizing constants.