  Importance sampling of target probability distributions belonging to a given
convex class is considered. Motivated by previous results, the cost of
importance sampling is quantified using the relative entropy of the target with
respect to proposal distributions. Using a reference measure as a reference for
cost, we prove under some general conditions that the worst-case optimal
proposal is precisely given by the distribution minimizing entropy with respect
to the reference within the considered convex class of distributions. The
latter conditions are in particular satisfied when the convex class is defined
using a push-forward map defining atomless conditional measures. Applications
in which the optimal proposal is Gibbsian and can be practically sampled using
Monte Carlo methods are discussed.
