# Amortized Bayesian Multilevel Models

arXiv: https://arxiv.org/abs/2408.13230 · HTML: https://mlsanity.com/paper/2408.13230/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 60 of its 117 references that are in the map.
- First posted on arXiv 23 Aug 2024.

## Builds on (most cited first, 30 of 60)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Neural Approximate Sufficient Statistics for Implicit Models](https://mlsanity.com/paper/2010.10079/) (2020)
- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Simulation Intelligence: Towards a New Generation of Scientific Methods](https://mlsanity.com/paper/2112.03235/) (2021)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Bayesian Workflow](https://mlsanity.com/paper/2011.01808/) (2020)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks](https://mlsanity.com/paper/2112.08866/) (2021)
- [Compositional Score Modeling for Simulation-based Inference](https://mlsanity.com/paper/2209.14249/) (2022)
- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)

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

- [Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data](https://mlsanity.com/paper/2501.13483/) (2025)
- [BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python](https://mlsanity.com/paper/2602.07098/) (2026)
- [Improving the Accuracy of Amortized Model Comparison with Self-Consistency](https://mlsanity.com/paper/2508.20614/) (2025)
- [Amortized Bayesian Mixture Models](https://mlsanity.com/paper/2501.10229/) (2025)
- [metabeta -- A fast neural model for Bayesian mixed-effects regression](https://mlsanity.com/paper/2510.07473/) (2025)
- [From Mice to Trains: Amortized Bayesian Inference on Graph Data](https://mlsanity.com/paper/2601.02241/) (2026)
- [Tokenised Flow Matching for Hierarchical Simulation Based Inference](https://mlsanity.com/paper/2604.20723/) (2026)
- [Amortized Bayesian Inference on Multilevel Models of Arbitrary Structure](https://mlsanity.com/paper/2609.40024/) (2026)
- [Improving the Accuracy of Amortized Model Comparison with Self-Consistency](https://mlsanity.com/paper/2512.14308/) (2025)
- [Unsupervised Continual Learning for Amortized Bayesian Inference](https://mlsanity.com/paper/2602.22884/) (2026)
- [Mixed neural posterior estimation for simulators with discrete and continuous parameters](https://mlsanity.com/paper/2605.13551/) (2026)
- [High-Dimensional Simulation-Based Inference in Latent Spaces](https://mlsanity.com/paper/2609.37381/) (2026)
- [Sample-Size Extrapolation in Simulation-Based Inference via Statistical Scaling](https://mlsanity.com/paper/2610.16820/) (2026)
- [Semi-Amortized Bayesian Inference with Reusable Local Posteriors](https://mlsanity.com/paper/2610.18945/) (2026)

## Abstract (arXiv)

Multilevel models (MLMs) are a central building block of the Bayesian workflow. They enable joint, interpretable modeling of data across hierarchical levels and provide a fully probabilistic quantification of uncertainty. Despite their well-recognized advantages, MLMs pose significant computational challenges, often rendering their estimation and evaluation intractable within reasonable time constraints. Recent advances in simulation-based inference offer promising solutions for addressing complex probabilistic models using deep generative networks. However, the utility and reliability of deep learning methods for estimating Bayesian MLMs remains largely unexplored, especially when compared with gold-standard samplers. To this end, we explore a family of neural network architectures that leverage the probabilistic factorization of multilevel models to facilitate efficient neural network training and subsequent near-instant posterior inference on unseen datasets. We test our method on several real-world case studies and provide comprehensive comparisons to Stan's gold standard sampler, where possible. Finally, we provide an open-source implementation of our methods to stimulate further research in the nascent field of amortized Bayesian inference.