Asymptotic Error Analysis of the MBAR Equations

The efficient estimation of high-dimensional integrals is central to answering many questions in statistical mechanics. An important class of sampling strategies, including umbrella sampling and alchemical methods, involves sampling from multiple thermodynamic states. A statistically optimal estimator named Multistate Bennett Acceptance Ratio (MBAR) has been developed to systematically combine data from all the sampled states to estimate free energies and other ensemble averages over another probability distribution. However, the error of the MBAR estimator is not well-understood: previous error analysis of MBAR assumed independent samples and only gave the total error without tracing the error to individual sampled states. In this work, We derive a central limit theorem for the estimates from MBAR, which allows for error decomposition into individual sampled states. We demonstrate the error estimator for a two-dimensional umbrella sampling of the alanine dipeptide and the alchemical free energy calculation of the solvation of methane in water. Our error analysis suggests a close connection between error contribution of each state and the autocorrelation in the samples from that state. The preliminary results suggest that assessing the error contributions provides insight into the sources of error of the simulations and can be used to improve the accuracy of MBAR calculations.