  Importance sampling approximates expectations with respect to a target
measure by using samples from a proposal measure. The performance of the method
over large classes of test functions depends heavily on the closeness between
both measures. We derive a general bound that needs to hold for importance
sampling to be successful, and relates the $f$-divergence between the target
and the proposal to the sample size. The bound is deduced from a new and simple
information theory paradigm for the study of importance sampling. As examples
of the general theory we give necessary conditions on the sample size in terms
of the Kullback-Leibler and $\chi^2$ divergences, and the total variation and
Hellinger distances. Our approach is non-asymptotic, and its generality allows
to tell apart the relative merits of these metrics. Unsurprisingly, the
non-symmetric divergences give sharper bounds than total variation or
Hellinger. Our results extend existing necessary conditions -and complement
sufficient ones- on the sample size required for importance sampling.
