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We present a new estimator for computing free energy differences andthermodynamic expectations as well as their uncertainties from samples obtainedfrom multiple equilibrium states via either simulation or experiment. Theestimator, which we term the multistate Bennett acceptance ratio (MBAR)estimator because it reduces to the Bennett acceptance ratio when only twostates are considered, has significant advantages over multiple histogramreweighting methods for combining data from multiple states. It does notrequire the sampled energy range to be discretized to produce histograms,eliminating bias due to energy binning and significantly reducing the timecomplexity of computing a solution to the estimating equations in many cases.Additionally, an estimate of the statistical uncertainty is provided for allestimated quantities. In the large sample limit, MBAR is unbiased and has thelowest variance of any known estimator for making use of equilibrium datacollected from multiple states. We illustrate this method by producing a highlyprecise estimate of the potential of mean force for a DNA hairpin system,combining data from multiple optical tweezer measurements under constant forcebias.

Statistically optimal analysis of samples from multiple equilibrium states