# Training Neural Networks for Likelihood/Density Ratio Estimation

arXiv: https://arxiv.org/abs/1911.00405 · HTML: https://mlsanity.com/paper/1911.00405/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2019.

## Built on by (most cited first, 17 of 17)

- [Contrastive Learning Can Find An Optimal Basis For Approximately View-Invariant Functions](https://mlsanity.com/paper/2210.01883/) (2022)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [Learning Likelihood Ratios with Neural Network Classifiers](https://mlsanity.com/paper/2305.10500/) (2023)
- [Computing high-dimensional optimal transport by flow neural networks](https://mlsanity.com/paper/2305.11857/) (2023)
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- [Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis](https://mlsanity.com/paper/2207.10939/) (2022)
- [Neural network-based CUSUM for online change-point detection](https://mlsanity.com/paper/2210.17312/) (2022)
- [Approximate Data Deletion in Generative Models](https://mlsanity.com/paper/2206.14439/) (2022)
- [The Power of Log-Sum-Exp: Sequential Density Ratio Matrix Estimation for Speed-Accuracy Optimization](https://mlsanity.com/paper/2105.13636/) (2021)
- [Embed and Emulate: Contrastive representations for simulation-based inference](https://mlsanity.com/paper/2409.18402/) (2024)
- [Sequential Change Point Detection via Denoising Score Matching](https://mlsanity.com/paper/2501.12667/) (2025)
- [Toward Asymptotic Optimality: Sequential Unsupervised Regression of Density Ratio for Early Classification](https://mlsanity.com/paper/2302.09810/) (2023)
- [Online non-parametric likelihood-ratio estimation by Pearson-divergence functional minimization](https://mlsanity.com/paper/2311.01900/) (2023)
- [Data-Driven Estimation of Conditional Expectations, Application to Optimal Stopping and Reinforcement Learning](https://mlsanity.com/paper/2407.13189/) (2024)
- [Uncertainty separation via ensemble quantile regression](https://mlsanity.com/paper/2412.13738/) (2024)
- [Neural Total Variation Distance Estimators for Changepoint Detection in News Data](https://mlsanity.com/paper/2506.18764/) (2025)
- [Context-aware Bayesian Fusion for Multimodal Object Detection](https://mlsanity.com/paper/2610.18741/) (2026)

## Abstract (arXiv)

Various problems in Engineering and Statistics require the computation of the likelihood ratio function of two probability densities. In classical approaches the two densities are assumed known or to belong to some known parametric family. In a data-driven version we replace this requirement with the availability of data sampled from the densities of interest. For most well known problems in Detection and Hypothesis testing we develop solutions by providing neural network based estimates of the likelihood ratio or its transformations. This task necessitates the definition of proper optimizations which can be used for the training of the network. The main purpose of this work is to offer a simple and unified methodology for defining such optimization problems with guarantees that the solution is indeed the desired function. Our results are extended to cover estimates for likelihood ratios of conditional densities and estimates for statistics encountered in local approaches.