  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.
