  We consider the problem of estimating rare event probabilities, focusing on
systems whose evolution is governed by differential equations with uncertain
input parameters. If the system dynamics is expensive to compute, standard
sampling algorithms such as the Monte Carlo method may require infeasible
running times to accurately evaluate these probabilities. We propose an
importance sampling scheme (which we call BIMC) that relies on solving an
auxiliary, fictitious Bayesian inverse problem. The solution of the inverse
problem yields a posterior PDF, a local Gaussian approximation to which serves
as the importance sampling density. We apply BIMC to several problems and
demonstrate that it can lead to computational savings of several orders of
magnitude over the Monte Carlo method. We delineate conditions under which BIMC
is optimal, as well as conditions when it can fail to yield an effective IS
density.
