Probabilistic performance estimators for computational chemistry methods: the empirical cumulative distribution function of absolute errors

TL;DR

This paper introduces the use of empirical cumulative distribution functions of absolute errors for better benchmarking in computational chemistry, providing more informative error probabilities than traditional statistics.

Contribution

It proposes new error statistics based on the empirical CDF of unsigned errors, improving benchmarking and ranking of computational methods.

Findings

Empirical CDF-based error probabilities are more informative.

Traditional error statistics do not reliably predict prediction error amplitudes.

Standard errors of benchmarking statistics depend on dataset size.

Abstract

Benchmarking studies in computational chemistry use reference datasets to assess the accuracy of a method through error statistics. The commonly used error statistics, such as the mean signed and mean unsigned errors, do not inform end-users on the expected amplitude of prediction errors attached to these methods. We show that, the distributions of model errors being neither normal nor zero-centered, these error statistics cannot be used to infer prediction error probabilities. To overcome this limitation, we advocate for the use of more informative statistics, based on the empirical cumulative distribution function of unsigned errors, namely (1) the probability for a new calculation to have an absolute error below a chosen threshold, and (2) the maximal amplitude of errors one can expect with a chosen high confidence level. Those statistics are also shown to be well suited for benchmarking and ranking studies. Moreover, the standard error on all benchmarking statistics depends on the size of the reference dataset. Systematic publication of these standard errors would be very helpful to assess the statistical reliability of benchmarking conclusions.