Estimating statistical distributions using an integral identity

TL;DR

This paper introduces a new integral identity for unbiased estimation of statistical distributions, enhancing robustness and precision over previous methods, and applicable to various distributions in molecular simulations.

Contribution

The authors develop a novel integral identity for distribution estimation, generalize it to arbitrary ensembles, and improve the weighted histogram analysis method (WHAM) for molecular simulations.

Findings

More accurate distribution estimates than previous methods.

Applicable to potential energy, volume, radial, and joint distributions.

Improves robustness and generality of distribution calculations.

Abstract

We present an identity for an unbiased estimate of a general statistical distribution. The identity computes the distribution density from dividing a histogram sum over a local window by a correction factor from a mean-force integral, and the mean force can be evaluated as a configuration average. We show that the optimal window size is roughly the inverse of the local mean-force fluctuation. The new identity offers a more robust and precise estimate than a previous one by Adib and Jarzynski [J. Chem. Phys. 122, 014114, (2005)]. It also allows a straightforward generalization to an arbitrary ensemble and a joint distribution of multiple variables. Particularly we derive a mean-force enhanced version of the weighted histogram analysis method (WHAM). The method can be used to improve distributions computed from molecular simulations. We illustrate the use in computing a potential energy distribution, a volume distribution in a constant-pressure ensemble, a radial distribution function and a joint distribution of amino acid backbone dihedral angles.