Improving Bridge estimators via $f$-GAN

Bridge sampling is a powerful Monte Carlo method for estimating ratios of normalizing constants. Various methods have been introduced to improve its efficiency. These methods aim to increase the overlap between the densities by applying appropriate transformations to them without changing their normalizing constants. In this paper, we first give a new estimator of the asymptotic relative mean square error (RMSE) of the optimal Bridge estimator by equivalently estimating an $f$-divergence between the two densities. We then utilize this framework and propose $f$-GAN-Bridge estimator ($f$-GB) based on a bijective transformation that maps one density to the other. Such transformation is chosen to minimize a specific $f$-divergence between them using an $f$-GAN \citep{nowozin2016f}. We show it is equivalent to minimizing the asymptotic RMSE of the optimal Bridge estimator with respect to the densities. In other words, $f$-GB is optimal in the sense that asymptotically, it can achieve an RMSE lower than that achieved by Bridge estimators based on any transformed density within the class of densities generated by the candidate transformations. Numerical experiments show that $f$-GB outperforms existing methods in simulated and real-world examples. In addition, we discuss how Bridge estimators naturally arise from the problem of $f$-divergence estimation.