# Learning Likelihood Ratios with Neural Network Classifiers

arXiv: https://arxiv.org/abs/2305.10500 · HTML: https://mlsanity.com/paper/2305.10500/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 60 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Approximating Likelihood Ratios with Calibrated Discriminative Classifiers](https://mlsanity.com/paper/1506.02169/) (2015)
- [Anomaly Detection with Density Estimation](https://mlsanity.com/paper/2001.04990/) (2020)
- [MadMiner: Machine learning-based inference for particle physics](https://mlsanity.com/paper/1907.10621/) (2019)
- [OmniFold: A Method to Simultaneously Unfold All Observables](https://mlsanity.com/paper/1911.09107/) (2019)
- [Simulation Assisted Likelihood-free Anomaly Detection](https://mlsanity.com/paper/2001.05001/) (2020)
- [Constraining Effective Field Theories with Machine Learning](https://mlsanity.com/paper/1805.00013/) (2018)
- [A Guide to Constraining Effective Field Theories with Machine Learning](https://mlsanity.com/paper/1805.00020/) (2018)
- [DCTRGAN: Improving the Precision of Generative Models with Reweighting](https://mlsanity.com/paper/2009.03796/) (2020)
- [Neural Networks for Full Phase-space Reweighting and Parameter Tuning](https://mlsanity.com/paper/1907.08209/) (2019)
- [JUNIPR: a Framework for Unsupervised Machine Learning in Particle Physics](https://mlsanity.com/paper/1804.09720/) (2018)
- [Training Neural Networks for Likelihood/Density Ratio Estimation](https://mlsanity.com/paper/1911.00405/) (2019)
- [E Pluribus Unum Ex Machina: Learning from Many Collider Events at Once](https://mlsanity.com/paper/2101.07263/) (2021)
- [Likelihood-free inference with an improved cross-entropy estimator](https://mlsanity.com/paper/1808.00973/) (2018)
- [Reweighting with Boosted Decision Trees](https://mlsanity.com/paper/1608.05806/) (2016)
- [Contrastive Neural Ratio Estimation for Simulation-based Inference](https://mlsanity.com/paper/2210.06170/) (2022)
- [Parameter Estimation using Neural Networks in the Presence of Detector Effects](https://mlsanity.com/paper/2010.03569/) (2020)
- [Flow Away your Differences: Conditional Normalizing Flows as an Improvement to Reweighting](https://mlsanity.com/paper/2304.14963/) (2023)

## Built on by (most cited first, 11 of 11)

- [Evidence Networks: simple losses for fast, amortized, neural Bayesian model comparison](https://mlsanity.com/paper/2305.11241/) (2023)
- [Optimal Equivariant Architectures from the Symmetries of Matrix-Element Likelihoods](https://mlsanity.com/paper/2410.18553/) (2024)
- [Constraining the Higgs Potential with Neural Simulation-based Inference for Di-Higgs Production](https://mlsanity.com/paper/2405.15847/) (2024)
- [Uncertainty-Aware Likelihood Ratio Estimation for Pixel-Wise Out-of-Distribution Detection](https://mlsanity.com/paper/2508.00587/) (2025)
- [Extrapolating Jet Radiation with Autoregressive Transformers](https://mlsanity.com/paper/2412.12074/) (2024)
- [Classification under Nuisance Parameters and Generalized Label Shift in Likelihood-Free Inference](https://mlsanity.com/paper/2402.05330/) (2024)
- [BitHEP -- The Limits of Low-Precision ML in HEP](https://mlsanity.com/paper/2504.03387/) (2025)
- [Temper-Then-Tilt: Principled Unlearning for Generative Models through Tempering and Classifier Guidance](https://mlsanity.com/paper/2602.10217/) (2026)
- [Conditional Neural Bayes Ratio Estimation for Experimental Design Optimisation](https://mlsanity.com/paper/2603.26489/) (2026)
- [An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning](https://mlsanity.com/paper/2607.21702/) (2026)
- [Quasiprobabilistic Density Ratio Estimation with a Reverse Engineered Classification Loss Function](https://mlsanity.com/paper/2512.19913/) (2025)

## Abstract (arXiv)

The likelihood ratio is a crucial quantity for statistical inference in science that enables hypothesis testing, construction of confidence intervals, reweighting of distributions, and more. Many modern scientific applications, however, make use of data- or simulation-driven models for which computing the likelihood ratio can be very difficult or even impossible. By applying the so-called ``likelihood ratio trick,'' approximations of the likelihood ratio may be computed using clever parametrizations of neural network-based classifiers. A number of different neural network setups can be defined to satisfy this procedure, each with varying performance in approximating the likelihood ratio when using finite training data. We present a series of empirical studies detailing the performance of several common loss functionals and parametrizations of the classifier output in approximating the likelihood ratio of two univariate and multivariate Gaussian distributions as well as simulated high-energy particle physics datasets.