Entropy minimizing distributions are worst-case optimal importance proposals

TL;DR

This paper shows that in importance sampling, the worst-case optimal proposal distribution within a convex class is the one that minimizes entropy relative to a reference measure, with practical applications in Monte Carlo methods.

Contribution

It establishes that entropy-minimizing distributions are the worst-case optimal importance proposals under general conditions, extending previous results.

Findings

Worst-case optimal proposals are entropy-minimizing distributions.

The results apply to convex classes defined via push-forward maps.

Practical sampling methods like Monte Carlo can implement these proposals.

Abstract

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.