Statistical uncertainty analysis for small-sample, high log-variance data: Cautions for bootstrapping and Bayesian bootstrapping

TL;DR

This paper examines the limitations of bootstrapping methods for estimating uncertainty in small, high log-variance data, highlighting biases and recommending Bayesian bootstrap with caution for more reliable confidence intervals.

Contribution

It demonstrates the biases of standard bootstrap in high log-variance data and advocates for Bayesian bootstrap as a more consistent alternative, with detailed analysis and practical insights.

Findings

Standard bootstrap exhibits systematic bias in high log-variance data.

Bayesian bootstrap provides more reliable uncertainty intervals than standard bootstrap.

Both methods struggle with small sample sizes and high log-variance, underestimating the mean.

Abstract

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 under-estimating the mean from small-size, high log-variance samples. Our conclusions are based on extensive analysis of model distributions and re-analysis of multiple independent atomistic simulations. Although we only analyze rate constants, similar considerations will apply to related calculations, potentially including highly non-linear averages like the Jarzynski relation.