  One popular approach to incorporating experimental data into molecular
simulations is to restrain the ensemble average of observables to their
experimental values. Here I derive equations for the equilibrium distributions
generated by restrained ensemble simulations and the corresponding expected
values of observables. My results suggest a method to restrain simulations so
that they generate distributions that are minimally perturbed from the unbiased
distributions while reproducing the experimental values of the observables
within their measurement uncertainties.
