Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence

TL;DR

This paper introduces a novel variational refinement method for importance sampling that minimizes the forward KL divergence, ensuring better tail behavior and asymptotic consistency in Bayesian inference.

Contribution

It proposes combining variational inference with importance sampling by minimizing the forward KL divergence, improving proposal quality and convergence.

Findings

Method is competitive with variational boosting.

Method achieves asymptotic consistency.

Demonstrated effectiveness on real data.

Abstract

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.