  The assessment of the probability of a rare event with a naive Monte-Carlo
method is computationally intensive, so faster estimation or variance reduction
methods are needed. We focus on one of these methods which is the interacting
particle system (IPS) method. The method is not intrusive in the sense that the
random Markov system under consideration is simulated with its original
distribution, but selection steps are introduced that favor trajectories
(particles) with high potential values. An unbiased estimator with reduced
variance can then be proposed. The method requires to specify a set of
potential functions. The choice of these functions is crucial, because it
determines the magnitude of the variance reduction. So far, little information
was available on how to choose the potential functions. This paper provides the
expressions of the optimal potential functions minimizing the asymptotic
variance of the estimator of the IPS method and it proposes recommendations for
the practical design of the potential functions.
