# Group equivariant neural posterior estimation

arXiv: https://arxiv.org/abs/2111.13139 · HTML: https://mlsanity.com/paper/2111.13139/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 51 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 23 of 23)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Real-time gravitational-wave science with neural posterior estimation](https://mlsanity.com/paper/2106.12594/) (2021)
- [Neural Approximate Sufficient Statistics for Implicit Models](https://mlsanity.com/paper/2010.10079/) (2020)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Learning Summary Statistic for Approximate Bayesian Computation via Deep Neural Network](https://mlsanity.com/paper/1510.02175/) (2015)
- [Bayesian parameter estimation using conditional variational autoencoders for gravitational-wave astronomy](https://mlsanity.com/paper/1909.06296/) (2019)
- [Complete parameter inference for GW150914 using deep learning](https://mlsanity.com/paper/2008.03312/) (2020)
- [Etalumis: Bringing Probabilistic Programming to Scientific Simulators at Scale](https://mlsanity.com/paper/1907.03382/) (2019)
- [Learning Bayesian posteriors with neural networks for gravitational-wave inference](https://mlsanity.com/paper/1909.05966/) (2019)
- [Lightning-Fast Gravitational Wave Parameter Inference through Neural Amortization](https://mlsanity.com/paper/2010.12931/) (2020)
- [Likelihood-free MCMC with Amortized Approximate Ratio Estimators](https://mlsanity.com/paper/1903.04057/) (2019)

## Built on by (most cited first, 16 of 16)

- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
- [Truncated proposals for scalable and hassle-free simulation-based inference](https://mlsanity.com/paper/2210.04815/) (2022)
- [Simulation-Based Inference: A Practical Guide](https://mlsanity.com/paper/2508.12939/) (2025)
- [Neural Importance Sampling for Rapid and Reliable Gravitational-Wave Inference](https://mlsanity.com/paper/2210.05686/) (2022)
- [Symmetry Group Equivariant Architectures for Physics](https://mlsanity.com/paper/2203.06153/) (2022)
- [Real-time gravitational-wave inference for binary neutron stars using machine learning](https://mlsanity.com/paper/2407.09602/) (2024)
- [L-C2ST: Local Diagnostics for Posterior Approximations in Simulation-Based Inference](https://mlsanity.com/paper/2306.03580/) (2023)
- [Effortless, Simulation-Efficient Bayesian Inference using Tabular Foundation Models](https://mlsanity.com/paper/2504.17660/) (2025)
- [Simultaneous identification of models and parameters of scientific simulators](https://mlsanity.com/paper/2305.15174/) (2023)
- [Inference-optimized AI and high performance computing for gravitational wave detection at scale](https://mlsanity.com/paper/2201.11133/) (2022)
- [Adapting to noise distribution shifts in flow-based gravitational-wave inference](https://mlsanity.com/paper/2211.08801/) (2022)
- [Fast and Reliable Probabilistic Reflectometry Inversion with Prior-Amortized Neural Posterior Estimation](https://mlsanity.com/paper/2407.18648/) (2024)
- [AI and extreme scale computing to learn and infer the physics of higher order gravitational wave modes of quasi-circular, spinning, non-precessing binary black hole mergers](https://mlsanity.com/paper/2112.07669/) (2021)
- [Flexible Gravitational-Wave Parameter Estimation with Transformers](https://mlsanity.com/paper/2512.02968/) (2025)
- [Fast, accurate, and differentiable: a neural-network surrogate for NRSur7dq4 precessing binary black hole waveforms](https://mlsanity.com/paper/2607.24960/) (2026)

## Abstract (arXiv)

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.