TY - JOUR

T1 - Statistical Uncertainty Analysis for Small-Sample, High Log-Variance Data

T2 - Cautions for Bootstrapping and Bayesian Bootstrapping

AU - Mostofian, Barmak

AU - Zuckerman, Daniel

PY - 2019/6/11

Y1 - 2019/6/11

N2 - Recent advances in molecular simulations allow the evaluation of previously unattainable observables, such as rate constants for protein folding. However, these calculations are usually computationally expensive, and even significant computing resources may result in a small number of independent estimates spread over many orders of magnitude. Such small-sample, high "log-variance" data are not readily amenable to analysis using the standard uncertainty (i.e., "standard error of the mean") because unphysical negative limits of confidence intervals result. Bootstrapping, a natural alternative guaranteed to yield a confidence interval within the minimum and maximum values, also exhibits a striking systematic bias of the lower confidence limit in log space. As we show, bootstrapping artifactually assigns high probability to improbably low mean values. A second alternative, the Bayesian bootstrap strategy, does not suffer from the same deficit and is more logically consistent with the type of confidence interval desired. The Bayesian bootstrap provides uncertainty intervals that are more reliable than those from the standard bootstrap method but must be used with caution nevertheless. Neither standard nor Bayesian bootstrapping can overcome the intrinsic challenge of underestimating the mean from small-size, high log-variance samples. Our conclusions are based on extensive analysis of model distributions and reanalysis of multiple independent atomistic simulations. Although we only analyze rate constants, similar considerations will apply to related calculations, potentially including highly nonlinear averages like the Jarzynski relation.

AB - Recent advances in molecular simulations allow the evaluation of previously unattainable observables, such as rate constants for protein folding. However, these calculations are usually computationally expensive, and even significant computing resources may result in a small number of independent estimates spread over many orders of magnitude. Such small-sample, high "log-variance" data are not readily amenable to analysis using the standard uncertainty (i.e., "standard error of the mean") because unphysical negative limits of confidence intervals result. Bootstrapping, a natural alternative guaranteed to yield a confidence interval within the minimum and maximum values, also exhibits a striking systematic bias of the lower confidence limit in log space. As we show, bootstrapping artifactually assigns high probability to improbably low mean values. A second alternative, the Bayesian bootstrap strategy, does not suffer from the same deficit and is more logically consistent with the type of confidence interval desired. The Bayesian bootstrap provides uncertainty intervals that are more reliable than those from the standard bootstrap method but must be used with caution nevertheless. Neither standard nor Bayesian bootstrapping can overcome the intrinsic challenge of underestimating the mean from small-size, high log-variance samples. Our conclusions are based on extensive analysis of model distributions and reanalysis of multiple independent atomistic simulations. Although we only analyze rate constants, similar considerations will apply to related calculations, potentially including highly nonlinear averages like the Jarzynski relation.

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U2 - 10.1021/acs.jctc.9b00015

DO - 10.1021/acs.jctc.9b00015

M3 - Article

C2 - 31002504

AN - SCOPUS:85066904292

VL - 15

SP - 3499

EP - 3509

JO - Journal of Chemical Theory and Computation

JF - Journal of Chemical Theory and Computation

SN - 1549-9618

IS - 6

ER -