# Likelihood-Free Parameter Estimation with Neural Bayes Estimators

arXiv: https://arxiv.org/abs/2208.12942 · HTML: https://mlsanity.com/paper/2208.12942/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 78 references that are in the map.
- First posted on arXiv 27 Aug 2022.

## Builds on (most cited first, 21 of 21)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs](https://mlsanity.com/paper/1811.01900/) (2018)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [Universal Approximation of Functions on Sets](https://mlsanity.com/paper/2107.01959/) (2021)
- [A Likelihood-Free Inference Framework for Population Genetic Data using Exchangeable Neural Networks](https://mlsanity.com/paper/1802.06153/) (2018)
- [Neural Networks for Parameter Estimation in Intractable Models](https://mlsanity.com/paper/2107.14346/) (2021)
- [Fast covariance parameter estimation of spatial Gaussian process models using neural networks](https://mlsanity.com/paper/2012.15339/) (2020)
- [Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE](https://mlsanity.com/paper/2012.06691/) (2020)
- [Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting](https://mlsanity.com/paper/1910.13524/) (2019)
- [Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2205.15784/) (2022)
- [On Deep Set Learning and the Choice of Aggregations](https://mlsanity.com/paper/1903.07348/) (2019)
- [Advances in Statistical Modeling of Spatial Extremes](https://mlsanity.com/paper/2007.00774/) (2020)
- [Full likelihood inference for max-stable data](https://mlsanity.com/paper/1703.08665/) (2017)
- [Higher-dimensional spatial extremes via single-site conditioning](https://mlsanity.com/paper/1912.06560/) (2019)
- [Modelling Extremes of Spatial Aggregates of Precipitation using Conditional Methods](https://mlsanity.com/paper/2102.10906/) (2021)
- [Conditional Modelling of Spatio-Temporal Extremes for Red Sea Surface Temperatures](https://mlsanity.com/paper/2002.04362/) (2020)
- [Vecchia Likelihood Approximation for Accurate and Fast Inference in Intractable Spatial Extremes Models](https://mlsanity.com/paper/2203.05626/) (2022)
- [High-dimensional modeling of spatial and spatio-temporal conditional extremes using INLA and Gaussian Markov random fields](https://mlsanity.com/paper/2011.04486/) (2020)

## Built on by (most cited first, 30 of 32)

- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods](https://mlsanity.com/paper/2305.04634/) (2023)
- [BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python](https://mlsanity.com/paper/2602.07098/) (2026)
- [Fast parameter estimation of Generalized Extreme Value distribution using Neural Networks](https://mlsanity.com/paper/2305.04341/) (2023)
- [Towards black-box parameter estimation](https://mlsanity.com/paper/2303.15041/) (2023)
- [Neural Bayes estimators for censored inference with peaks-over-threshold models](https://mlsanity.com/paper/2306.15642/) (2023)
- [Paired Autoencoders for Likelihood-free Estimation in Inverse Problems](https://mlsanity.com/paper/2405.13220/) (2024)
- [Generative Bayesian Computation for Maximum Expected Utility](https://mlsanity.com/paper/2408.16101/) (2024)
- [Deep Optimal Experimental Design for Parameter Estimation Problems](https://mlsanity.com/paper/2406.14003/) (2024)
- [Deep Generative Quantile Bayes](https://mlsanity.com/paper/2410.08378/) (2024)
- [Theoretical guarantees for neural estimators in parametric statistics](https://mlsanity.com/paper/2506.18508/) (2025)
- [Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks](https://mlsanity.com/paper/2505.03034/) (2025)
- [LatticeVision: Image to Image Networks for Modeling Non-Stationary Spatial Data](https://mlsanity.com/paper/2505.09803/) (2025)
- [Trustworthy scientific inference with generative models](https://mlsanity.com/paper/2508.02602/) (2025)
- [It Just Takes Two: Scaling Amortized Inference to Large Sets](https://mlsanity.com/paper/2605.07972/) (2026)
- [Latent Variable Sequence Identification for Cognitive Models with Neural Network Estimators](https://mlsanity.com/paper/2406.14742/) (2024)
- [Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks](https://mlsanity.com/paper/2407.08668/) (2024)
- [A Generalized Unified Skew-Normal Process with Neural Bayes Inference](https://mlsanity.com/paper/2411.17400/) (2024)
- [When the whole is greater than the sum of its parts: Scaling black-box inference to large data settings through divide-and-conquer](https://mlsanity.com/paper/2412.20323/) (2024)
- [Neural Bayes estimation and selection for complex bivariate extremal dependence models](https://mlsanity.com/paper/2503.23156/) (2025)
- [Fast Likelihood-Free Parameter Estimation for L'evy Processes](https://mlsanity.com/paper/2505.01639/) (2025)
- [Neural Architectures for Amortized Bayesian Inference: Statistical Foundations and Empirical Assessments](https://mlsanity.com/paper/2601.07944/) (2026)
- [Amortized Bayesian inference for actigraph time sheet data from mobile devices](https://mlsanity.com/paper/2602.20611/) (2026)
- [Permutation-preserving Functions and Neural Vecchia Covariance Kernels](https://mlsanity.com/paper/2605.05523/) (2026)
- [Keeping Score: Adaptive, Tuning-Free Loss Weighting for Score-Augmented Neural Ratio Estimation](https://mlsanity.com/paper/2605.12118/) (2026)
- [Benchmark of Likelihood-Free Inference Methods based on Neural and Optimal Transport Approaches](https://mlsanity.com/paper/2605.30516/) (2026)
- [The Simulacrum: Decision-Theoretic Pretraining for Near-Optimal Time-Series Forecasting and Inference](https://mlsanity.com/paper/2606.27711/) (2026)
- [Stochastic generator of trajectories from record data: application to the fluctuations of a glacier's frontal position from a sample of moraines](https://mlsanity.com/paper/2607.06020/) (2026)
- [Amortized Inference for Sampling Distributions Where the Bootstrap Fails](https://mlsanity.com/paper/2607.16666/) (2026)

## Abstract (arXiv)

Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second.