The paper presents the multistate Bennett acceptance ratio (MBAR) estimator, a statistically optimal method for computing free energy differences and thermodynamic expectations from samples obtained from multiple equilibrium states. Unlike multiple histogram reweighting methods, MBAR does not require discretization of energy ranges, eliminating bias and reducing computational complexity. It provides direct estimates of statistical uncertainties and is asymptotically unbiased with the lowest variance among known estimators. The method is illustrated by producing a highly precise estimate of the potential of mean force for a DNA hairpin system using data from multiple optical tweezer measurements. MBAR is applicable to both simulation and experimental data, and can handle non-Boltzmann sampling schemes and single-molecule experiments with external bias potentials. The estimator is derived from statistical inference principles and is equivalent to the weighted histogram analysis method (WHAM) in the limit of zero histogram bin widths. It provides a computationally efficient way to estimate free energy differences and equilibrium expectations, with the ability to handle large numbers of states. The paper also discusses the application of MBAR to laboratory experiments, demonstrating its effectiveness in estimating the potential of mean force for a DNA hairpin system under different external biasing potentials. The method is shown to produce significantly smaller error bars compared to traditional methods, and is robust and efficient for a wide range of applications.The paper presents the multistate Bennett acceptance ratio (MBAR) estimator, a statistically optimal method for computing free energy differences and thermodynamic expectations from samples obtained from multiple equilibrium states. Unlike multiple histogram reweighting methods, MBAR does not require discretization of energy ranges, eliminating bias and reducing computational complexity. It provides direct estimates of statistical uncertainties and is asymptotically unbiased with the lowest variance among known estimators. The method is illustrated by producing a highly precise estimate of the potential of mean force for a DNA hairpin system using data from multiple optical tweezer measurements. MBAR is applicable to both simulation and experimental data, and can handle non-Boltzmann sampling schemes and single-molecule experiments with external bias potentials. The estimator is derived from statistical inference principles and is equivalent to the weighted histogram analysis method (WHAM) in the limit of zero histogram bin widths. It provides a computationally efficient way to estimate free energy differences and equilibrium expectations, with the ability to handle large numbers of states. The paper also discusses the application of MBAR to laboratory experiments, demonstrating its effectiveness in estimating the potential of mean force for a DNA hairpin system under different external biasing potentials. The method is shown to produce significantly smaller error bars compared to traditional methods, and is robust and efficient for a wide range of applications.