  Importance Sampling (IS) is a method for approximating expectations under a
target distribution using independent samples from a proposal distribution and
the associated importance weights. In many applications, the target
distribution is known only up to a normalization constant, in which case
self-normalized IS (SNIS) can be used. While the use of self-normalization can
have a positive effect on the dispersion of the estimator, it introduces bias.
In this work, we propose a new method, BR-SNIS, whose complexity is essentially
the same as that of SNIS and which significantly reduces bias without
increasing the variance. This method is a wrapper in the sense that it uses the
same proposal samples and importance weights as SNIS, but makes clever use of
iterated sampling--importance resampling (ISIR) to form a bias-reduced version
of the estimator. We furnish the proposed algorithm with rigorous theoretical
results, including new bias, variance and high-probability bounds, and these
are illustrated by numerical examples.
