Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages

TL;DR

This paper introduces a unified framework for optimal estimators of nonequilibrium path-ensemble averages, providing minimal-variance estimators and asymptotic variance estimates applicable to various free energy calculations, validated through a single-molecule pulling experiment.

Contribution

It develops a general minimal-variance estimator within the extended bridge sampling framework for combining multiple nonequilibrium trajectory data sets.

Findings

The asymptotic variance estimates accurately characterize confidence intervals with small bias.

The framework applies to Jarzynski's equality and potential of mean force calculations.

Demonstrated effectiveness on a model single-molecule pulling experiment.

Abstract

Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtaining asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. Lastly, they are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, it does not work well for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.