# Amortized Bayesian Inference on Multilevel Models of Arbitrary Structure

arXiv: https://arxiv.org/abs/2609.40024 · HTML: https://mlsanity.com/paper/2609.40024/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Transformers in Time Series: A Survey](https://mlsanity.com/paper/2202.07125/) (2022)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [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)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [BayesFlow: Amortized Bayesian Workflows With Neural Networks](https://mlsanity.com/paper/2306.16015/) (2023)
- [Faithful Inversion of Generative Models for Effective Amortized Inference](https://mlsanity.com/paper/1712.00287/) (2017)
- [Visualization in Bayesian workflow](https://mlsanity.com/paper/1709.01449/) (2017)
- [Prior knowledge elicitation: The past, present, and future](https://mlsanity.com/paper/2112.01380/) (2021)
- [Real-time gravitational-wave inference for binary neutron stars using machine learning](https://mlsanity.com/paper/2407.09602/) (2024)
- [HNPE: Leveraging Global Parameters for Neural Posterior Estimation](https://mlsanity.com/paper/2102.06477/) (2021)
- [Simulation-Based Calibration Checking for Bayesian Computation: The Choice of Test Quantities Shapes Sensitivity](https://mlsanity.com/paper/2211.02383/) (2022)
- [OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and its application to the COVID-19 pandemics in Germany](https://mlsanity.com/paper/2010.00300/) (2020)
- [Amortized Bayesian Multilevel Models](https://mlsanity.com/paper/2408.13230/) (2024)
- [Simulation-based Inference for Cardiovascular Models](https://mlsanity.com/paper/2307.13918/) (2023)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)
- [Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data](https://mlsanity.com/paper/2501.13483/) (2025)
- [Hierarchical Neural Simulation-Based Inference Over Event Ensembles](https://mlsanity.com/paper/2306.12584/) (2023)
- [BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python](https://mlsanity.com/paper/2602.07098/) (2026)
- [Amortized Bayesian Mixture Models](https://mlsanity.com/paper/2501.10229/) (2025)
- [CogFormer: Learn All Your Models Once](https://mlsanity.com/paper/2603.20520/) (2026)

## Built on by (most cited first, 1 of 1)

- [Semi-Amortized Bayesian Inference with Reusable Local Posteriors](https://mlsanity.com/paper/2610.18945/) (2026)

## Abstract (arXiv)

We develop a general method for amortized Bayesian inference on multilevel models of arbitrary structure. Given a generative model specified as a directed acyclic graph, our method automatically derives valid factorizations of the joint posterior and matching neural network architectures. The key steps, graph expansion and graph inversion, yield an inverse graph that determines how inference networks are stacked and conditioned, producing factorizations that amortize over the number of groups and the number of observations within each group. Unlike approaches that simplify the dependency structure to speed up learning or inference, our method preserves all conditional independence and exchangeability assumptions of the generative model. Across three case studies, it closely matches gold-standard samplers on models with more than 6,500 parameters while reducing inference to a near-instant forward pass once trained.