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

arXiv: https://arxiv.org/abs/2106.15980 · HTML: https://mlsanity.com/paper/2106.15980/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 60 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Pareto Smoothed Importance Sampling](https://mlsanity.com/paper/1507.02646/) (2015)
- [Variational Boosting: Iteratively Refining Posterior Approximations](https://mlsanity.com/paper/1611.06585/) (2016)
- [Validated Variational Inference via Practical Posterior Error Bounds](https://mlsanity.com/paper/1910.04102/) (2019)
- [Boosting Variational Inference](https://mlsanity.com/paper/1611.05559/) (2016)
- [A Contrastive Divergence for Combining Variational Inference and MCMC](https://mlsanity.com/paper/1905.04062/) (2019)
- [Universal Boosting Variational Inference](https://mlsanity.com/paper/1906.01235/) (2019)
- [Boosting Variational Inference: an Optimization Perspective](https://mlsanity.com/paper/1708.01733/) (2017)
- [Irreversible Samplers from Jump and Continuous Markov Processes](https://mlsanity.com/paper/1608.05973/) (2016)

## Built on by (most cited first, 29 of 29)

- [Understanding Failures in Out-of-Distribution Detection with Deep Generative Models](https://mlsanity.com/paper/2107.06908/) (2021)
- [Variational methods for simulation-based inference](https://mlsanity.com/paper/2203.04176/) (2022)
- [Greedification Operators for Policy Optimization: Investigating Forward and Reverse KL Divergences](https://mlsanity.com/paper/2107.08285/) (2021)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [A theoretical perspective on mode collapse in variational inference](https://mlsanity.com/paper/2410.13300/) (2024)
- [Learned Reference-based Diffusion Sampling for multi-modal distributions](https://mlsanity.com/paper/2410.19449/) (2024)
- [Improving the evaluation of samplers on multi-modal targets](https://mlsanity.com/paper/2504.08916/) (2025)
- [Transport Score Climbing: Variational Inference Using Forward KL and Adaptive Neural Transport](https://mlsanity.com/paper/2202.01841/) (2022)
- [Bridging the Gap Between Variational Inference and Wasserstein Gradient Flows](https://mlsanity.com/paper/2310.20090/) (2023)
- [Markov Chain Score Ascent: A Unifying Framework of Variational Inference with Markovian Gradients](https://mlsanity.com/paper/2206.06295/) (2022)
- [Stable Training of Normalizing Flows for High-dimensional Variational Inference](https://mlsanity.com/paper/2402.16408/) (2024)
- [Balanced Training of Energy-Based Models with Adaptive Flow Sampling](https://mlsanity.com/paper/2306.00684/) (2023)
- [Forward chi2 Divergence Based Variational Importance Sampling](https://mlsanity.com/paper/2311.02516/) (2023)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [Variational Bayes for high-dimensional proportional hazards models with applications within gene expression](https://mlsanity.com/paper/2112.10270/) (2021)
- [Global convergence of optimized adaptive importance samplers](https://mlsanity.com/paper/2201.00409/) (2022)
- [Learning to Generate All Feasible Actions](https://mlsanity.com/paper/2301.11461/) (2023)
- [metabeta -- A fast neural model for Bayesian mixed-effects regression](https://mlsanity.com/paper/2510.07473/) (2025)
- [Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation](https://mlsanity.com/paper/2601.22367/) (2026)
- [SoftCVI: Contrastive variational inference with self-generated soft labels](https://mlsanity.com/paper/2407.15687/) (2024)
- [Safe Beyond the Horizon: Efficient Sampling-based MPC with Neural Control Barrier Functions](https://mlsanity.com/paper/2502.15006/) (2025)
- [How to Approximate Inference with Subtractive Mixture Models](https://mlsanity.com/paper/2604.16714/) (2026)
- [Even More Guarantees for Variational Inference in the Presence of Symmetries](https://mlsanity.com/paper/2604.21407/) (2026)
- [Stochastic automatic differentiation for Monte Carlo processes](https://mlsanity.com/paper/2307.15406/) (2023)
- [Two tales for a geometric Jensen--Shannon divergence](https://mlsanity.com/paper/2508.05066/) (2025)
- [TinyMusician: On-Device Music Generation with Knowledge Distillation and Mixed Precision Quantization](https://mlsanity.com/paper/2509.00914/) (2025)
- [Total Variation Guarantees for Sampling with Stochastic Localization](https://mlsanity.com/paper/2603.29555/) (2026)
- [Amortized Factor Inference Networks for Posterior Inference](https://mlsanity.com/paper/2605.26419/) (2026)

## Abstract (arXiv)

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.