  We revisit the work of Mitter and Newton on an information-theoretic
interpretation of Bayes' formula through the Gibbs variational principle. This
formulation allowed them to pose nonlinear estimation for diffusion processes
as a problem in stochastic optimal control, so that the posterior density of
the signal given the observation path could be sampled by adding a drift to the
signal process. We show that this control-theoretic approach to sampling
provides a common mechanism underlying several distinct problems involving
diffusion processes, specifically importance sampling using Feynman-Kac
averages, time reversal, and Schr\"odinger bridges.
