# Nested Variational Inference

arXiv: https://arxiv.org/abs/2106.11302 · HTML: https://mlsanity.com/paper/2106.11302/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 56 references that are in the map.
- First posted on arXiv 21 Jun 2021.

## Builds on (most cited first, 30 of 35)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Gradient Estimation Using Stochastic Computation Graphs](https://mlsanity.com/paper/1506.05254/) (2015)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Filtering Variational Objectives](https://mlsanity.com/paper/1705.09279/) (2017)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Auto-Encoding Sequential Monte Carlo](https://mlsanity.com/paper/1705.10306/) (2017)
- [Elements of Sequential Monte Carlo](https://mlsanity.com/paper/1903.04797/) (2019)
- [On Particle Methods for Parameter Estimation in State-Space Models](https://mlsanity.com/paper/1412.8695/) (2014)
- [Hamiltonian Variational Auto-Encoder](https://mlsanity.com/paper/1805.11328/) (2018)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [The continuous Bernoulli: fixing a pervasive error in variational autoencoders](https://mlsanity.com/paper/1907.06845/) (2019)
- [Automated Variational Inference in Probabilistic Programming](https://mlsanity.com/paper/1301.1299/) (2013)
- [The Thermodynamic Variational Objective](https://mlsanity.com/paper/1907.00031/) (2019)
- [Improving Explorability in Variational Inference with Annealed Variational Objectives](https://mlsanity.com/paper/1809.01818/) (2018)
- [Deep Amortized Inference for Probabilistic Programs](https://mlsanity.com/paper/1610.05735/) (2016)
- [Approximate Inference with Amortised MCMC](https://mlsanity.com/paper/1702.08343/) (2017)
- [A New Approach to Probabilistic Programming Inference](https://mlsanity.com/paper/1507.00996/) (2015)
- [Automatic structured variational inference](https://mlsanity.com/paper/2002.00643/) (2020)
- [Nested Sequential Monte Carlo Methods](https://mlsanity.com/paper/1502.02536/) (2015)
- [Variational Inference with Hamiltonian Monte Carlo](https://mlsanity.com/paper/1609.08203/) (2016)
- [Generalized Doubly Reparameterized Gradient Estimators](https://mlsanity.com/paper/2101.11046/) (2021)
- [Automated learning with a probabilistic programming language: Birch](https://mlsanity.com/paper/1810.01539/) (2018)
- [Amortized Population Gibbs Samplers with Neural Sufficient Statistics](https://mlsanity.com/paper/1911.01382/) (2019)
- [Interacting Particle Markov Chain Monte Carlo](https://mlsanity.com/paper/1602.05128/) (2016)

## Built on by (most cited first, 14 of 14)

- [GFlowNets and variational inference](https://mlsanity.com/paper/2210.00580/) (2022)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [A Variational Perspective on Generative Flow Networks](https://mlsanity.com/paper/2210.07992/) (2022)
- [Deep Generative Models for Detector Signature Simulation: A Taxonomic Review](https://mlsanity.com/paper/2312.09597/) (2023)
- [Recursive Monte Carlo and Variational Inference with Auxiliary Variables](https://mlsanity.com/paper/2203.02836/) (2022)
- [Bridge the Inference Gaps of Neural Processes via Expectation Maximization](https://mlsanity.com/paper/2501.03264/) (2025)
- [Sequential Monte Carlo for Inclusive KL Minimization in Amortized Variational Inference](https://mlsanity.com/paper/2403.10610/) (2024)
- [GFlowNet Training by Policy Gradients](https://mlsanity.com/paper/2408.05885/) (2024)
- [Black-box Coreset Variational Inference](https://mlsanity.com/paper/2211.02377/) (2022)
- [NAS-X: Neural Adaptive Smoothing via Twisting](https://mlsanity.com/paper/2308.14864/) (2023)
- [Probabilistic Programming with Programmable Variational Inference](https://mlsanity.com/paper/2406.15742/) (2024)
- [VISA: Variational Inference with Sequential Sample-Average Approximations](https://mlsanity.com/paper/2403.09429/) (2024)
- [Deep Generative Sampling in the Dual Divergence Space: A Data-efficient & Interpretative Approach for Generative AI](https://mlsanity.com/paper/2404.07377/) (2024)
- [Revisiting Unbiased Implicit Variational Inference](https://mlsanity.com/paper/2506.03839/) (2025)

## Abstract (arXiv)

We develop nested variational inference (NVI), a family of methods that learn proposals for nested importance samplers by minimizing an forward or reverse KL divergence at each level of nesting. NVI is applicable to many commonly-used importance sampling strategies and provides a mechanism for learning intermediate densities, which can serve as heuristics to guide the sampler. Our experiments apply NVI to (a) sample from a multimodal distribution using a learned annealing path (b) learn heuristics that approximate the likelihood of future observations in a hidden Markov model and (c) to perform amortized inference in hierarchical deep generative models. We observe that optimizing nested objectives leads to improved sample quality in terms of log average weight and effective sample size.