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.Comment: 45 pages, 7 figures, simplified derivation, a more general mean-force
formula, add discussions to the window size, add extensions to WHAM, and 2d
distribution
