  Variational Inference (VI) is a popular alternative to asymptotically exact
sampling in Bayesian inference. Its main workhorse is optimization over a
reverse Kullback-Leibler divergence (RKL), which typically underestimates the
tail of the posterior leading to miscalibration and potential degeneracy.
Importance sampling (IS), on the other hand, is often used to fine-tune and
de-bias the estimates of approximate Bayesian inference procedures. The quality
of IS crucially depends on the choice of the proposal distribution. Ideally,
the proposal distribution has heavier tails than the target, which is rarely
achievable by minimizing the RKL. We thus propose a novel combination of
optimization and sampling techniques for approximate Bayesian inference by
constructing an IS proposal distribution through the minimization of a forward
KL (FKL) divergence. This approach guarantees asymptotic consistency and a fast
convergence towards both the optimal IS estimator and the optimal variational
approximation. We empirically demonstrate on real data that our method is
competitive with variational boosting and MCMC.
