# Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization

arXiv: https://arxiv.org/abs/2205.15784 · HTML: https://mlsanity.com/paper/2205.15784/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 31 May 2022.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (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)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [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)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [On GANs and GMMs](https://mlsanity.com/paper/1805.12462/) (2018)
- [Likelihood-free inference with emulator networks](https://mlsanity.com/paper/1805.09294/) (2018)
- [Fast ε-free Inference of Simulation Models with Bayesian Conditional Density Estimation](https://mlsanity.com/paper/1605.06376/) (2016)
- [A Spectral Energy Distance for Parallel Speech Synthesis](https://mlsanity.com/paper/2008.01160/) (2020)
- [Estimating and Evaluating Regression Predictive Uncertainty in Deep Object Detectors](https://mlsanity.com/paper/2101.05036/) (2021)
- [GATSBI: Generative Adversarial Training for Simulation-Based Inference](https://mlsanity.com/paper/2203.06481/) (2022)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [Approximate Bayesian computation with the Wasserstein distance](https://mlsanity.com/paper/1905.03747/) (2019)
- [DISCO Nets: DISsimilarity COefficient Networks](https://mlsanity.com/paper/1606.02556/) (2016)
- [Robust Bayesian Synthetic Likelihood via a Semi-Parametric Approach](https://mlsanity.com/paper/1809.05800/) (2018)
- [Approximate Bayesian computation via the energy statistic](https://mlsanity.com/paper/1905.05884/) (2019)

## Built on by (most cited first, 15 of 15)

- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Continuous Visual Autoregressive Generation via Score Maximization](https://mlsanity.com/paper/2505.07812/) (2025)
- [Language Generation with Strictly Proper Scoring Rules](https://mlsanity.com/paper/2405.18906/) (2024)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks: An Extended Investigation](https://mlsanity.com/paper/2406.03154/) (2024)
- [Learning Probabilistic Filters with Strictly Proper Scoring Rules](https://mlsanity.com/paper/2606.26497/) (2026)
- [Probabilistic Self-supervised Learning via Scoring Rules Minimization](https://mlsanity.com/paper/2309.02048/) (2023)
- [Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions](https://mlsanity.com/paper/2410.03973/) (2024)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Bayesian score calibration for approximate models](https://mlsanity.com/paper/2211.05357/) (2022)
- [Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks](https://mlsanity.com/paper/2407.08668/) (2024)
- [Likelihood-Free Variational Autoencoders](https://mlsanity.com/paper/2504.17622/) (2025)

## Abstract (arXiv)

Bayesian Likelihood-Free Inference methods yield posterior approximations for simulator models with intractable likelihood. Recently, many works trained neural networks to approximate either the intractable likelihood or the posterior directly. Most proposals use normalizing flows, namely neural networks parametrizing invertible maps used to transform samples from an underlying base measure; the probability density of the transformed samples is then accessible and the normalizing flow can be trained via maximum likelihood on simulated parameter-observation pairs. A recent work [Ramesh et al., 2022] approximated instead the posterior with generative networks, which drop the invertibility requirement and are thus a more flexible class of distributions scaling to high-dimensional and structured data. However, generative networks only allow sampling from the parametrized distribution; for this reason, Ramesh et al. [2022] follows the common solution of adversarial training, where the generative network plays a min-max game against a "critic" network. This procedure is unstable and can lead to a learned distribution underestimating the uncertainty - in extreme cases collapsing to a single point. Here, we propose to approximate the posterior with generative networks trained by Scoring Rule minimization, an overlooked adversarial-free method enabling smooth training and better uncertainty quantification. In simulation studies, the Scoring Rule approach yields better performances with shorter training time with respect to the adversarial framework.