A theoretical interpretation of variance-based convergence criteria in perturbation-based theories

Abstract: In QM/MM indirect free energy simulation, QM/MM corrections can be obtained from integration of partial derivatives of alchemical Hamiltonians or from perturbation-based estimators including free energy perturbation (FEP) and acceptance ratio methods. With FEP or exponential averaging, researchers tend to only sample MM states and calculate single point energy to get the free energy estimates. In this case the sample size hysteresis arises and the convergence is determined by bias elimination rather than variance minimization. Various criteria are proposed to evaluate the convergence issue and numerical studies are reported. It has been found that criteria including variance of distribution, effective sample size, information entropies and so on can be used and they are variance-of-distribution-dependent. However, no theoretical interpretation is presented. In this paper we present theoretical interpretations to dig the underlying statistical nature behind the problem. The convergence criteria are proven to be related with variance of distribution in Gaussian approximated Exponential averaging. Further, we prove that these estimators are nonlinearly dependent on the variance of the free energy estimate. As these estimators are often orders of magnitude overestimated, the variance of the FEP estimate is orders of magnitude underestimated. Hence, computing this statistical uncertainty is meaningless. In numerical calculation from timeseries data the effective sample size is bounded by 1 and N and thus the variance of the free energy estimate is proven to be bounded by 0 and 1 (kBT)2 for EXP and 0 and 2 (kBT)2 for BAR, which indicates an inevitable underestimation. Specifically, the upper bounds for these estimators are sample-size dependent. The effective sample size is proven to be a function of the overlap scalar, from which the range of the overlap scalar can also be derived.