# Amortized Variational Inference: When and Why?

arXiv: https://arxiv.org/abs/2307.11018 · HTML: https://mlsanity.com/paper/2307.11018/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 20 Jul 2023.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Autoencoding Variational Inference For Topic Models](https://mlsanity.com/paper/1703.01488/) (2017)
- [Inference Suboptimality in Variational Autoencoders](https://mlsanity.com/paper/1801.03558/) (2018)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Iterative Amortized Inference](https://mlsanity.com/paper/1807.09356/) (2018)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [Amortized Inference Regularization](https://mlsanity.com/paper/1805.08913/) (2018)
- [Faithful Inversion of Generative Models for Effective Amortized Inference](https://mlsanity.com/paper/1712.00287/) (2017)
- [Automatic structured variational inference](https://mlsanity.com/paper/2002.00643/) (2020)
- [On the challenges of learning with inference networks on sparse, high-dimensional data](https://mlsanity.com/paper/1710.06085/) (2017)
- [Amortized Variational Inference for Simple Hierarchical Models](https://mlsanity.com/paper/2111.03144/) (2021)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [The Shrinkage-Delinkage Trade-off: An Analysis of Factorized Gaussian Approximations for Variational Inference](https://mlsanity.com/paper/2302.09163/) (2023)
- [Reducing the Amortization Gap in Variational Autoencoders: A Bayesian Random Function Approach](https://mlsanity.com/paper/2102.03151/) (2021)
- [Monte Carlo Structured SVI for Two-Level Non-Conjugate Models](https://mlsanity.com/paper/1612.03957/) (2016)

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

- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Adaptive teachers for amortized samplers](https://mlsanity.com/paper/2410.01432/) (2024)
- [Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs](https://mlsanity.com/paper/2403.13748/) (2024)
- [Intention-Conditioned Flow Occupancy Models](https://mlsanity.com/paper/2506.08902/) (2025)
- [Riemannian Laplace Approximation with the Fisher Metric](https://mlsanity.com/paper/2311.02766/) (2023)
- [Trust-Region Twisted Policy Improvement](https://mlsanity.com/paper/2504.06048/) (2025)
- [Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation](https://mlsanity.com/paper/2603.28744/) (2026)
- [Deep Active Inference Agents for Delayed and Long-Horizon Environments](https://mlsanity.com/paper/2505.19867/) (2025)
- [Diffusion Bridge Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/2509.19078/) (2025)
- [Meta-Statistical Learning: Supervised Learning of Statistical Estimators](https://mlsanity.com/paper/2502.12088/) (2025)
- [ComFe: An Interpretable Head for Vision Transformers](https://mlsanity.com/paper/2403.04125/) (2024)
- [Amortized Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/2409.12301/) (2024)
- [Sparse Gaussian Neural Processes](https://mlsanity.com/paper/2504.01650/) (2025)
- [Quantifying Uncertainty in the Presence of Distribution Shifts](https://mlsanity.com/paper/2506.18283/) (2025)
- [Active Inference Meeting Energy-Efficient Control of Parallel and Identical Machines](https://mlsanity.com/paper/2406.09322/) (2024)
- [A Frequentist Statistical Introduction to Variational Inference, Autoencoders, and Diffusion Models](https://mlsanity.com/paper/2510.18777/) (2025)
- [VariBASed: Variational Bayes-Adaptive Sequential Monte-Carlo Planning for Deep Reinforcement Learning](https://mlsanity.com/paper/2602.18857/) (2026)
- [Simulation-Based Empirical Bayes](https://mlsanity.com/paper/2607.21843/) (2026)
- [Slow Feature Analysis as Variational Inference Objective](https://mlsanity.com/paper/2506.00580/) (2025)
- [Misspecification-robust amortised simulation-based inference using variational methods](https://mlsanity.com/paper/2509.05724/) (2025)
- [Amortized Variational Inference for Partial-Label Learning: A Probabilistic Approach to Label Disambiguation](https://mlsanity.com/paper/2510.21300/) (2025)
- [Density-Informed Pseudo-Counts for Calibrated Evidential Deep Learning](https://mlsanity.com/paper/2602.01477/) (2026)
- [Amortized Variational Inference for Logistic Regression with Missing Covariates](https://mlsanity.com/paper/2603.21244/) (2026)
- [GLU: Global-Local-Uncertainty Fusion for Scalable Spatiotemporal Reconstruction and Forecasting](https://mlsanity.com/paper/2603.26023/) (2026)
- [A Decision-Theoretic View of Test-Time Training: When, How Far, and Which Directions to Adapt](https://mlsanity.com/paper/2606.15569/) (2026)
- [Learning Implicit Causal World Models from Multi-Agent Demonstrations](https://mlsanity.com/paper/2607.26336/) (2026)
- [Scalable Amortized Variational Inference for Non-Poisson Buy-'Til-You-Die Models](https://mlsanity.com/paper/2608.19022/) (2026)
- [Interference Beyond Geometry in Concept Extraction](https://mlsanity.com/paper/2609.35351/) (2026)
- [The Independence Prior of SAEs Fragments Visual Concepts](https://mlsanity.com/paper/2610.04112/) (2026)

## Abstract (arXiv)

In a probabilistic latent variable model, factorized (or mean-field) variational inference (F-VI) fits a separate parametric distribution for each latent variable. Amortized variational inference (A-VI) instead learns a common inference function, which maps each observation to its corresponding latent variable's approximate posterior. Typically, A-VI is used as a step in the training of variational autoencoders, however it stands to reason that A-VI could also be used as a general alternative to F-VI. In this paper we study when and why A-VI can be used for approximate Bayesian inference. We derive conditions on a latent variable model which are necessary, sufficient, and verifiable under which A-VI can attain F-VI's optimal solution, thereby closing the amortization gap. We prove these conditions are uniquely verified by simple hierarchical models, a broad class that encompasses many models in machine learning. We then show, on a broader class of models, how to expand the domain of AVI's inference function to improve its solution, and we provide examples, e.g. hidden Markov models, where the amortization gap cannot be closed.