# Contrastive Normalizing Flows for Uncertainty-Aware Parameter Estimation

arXiv: https://arxiv.org/abs/2505.08709 · HTML: https://mlsanity.com/paper/2505.08709/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 20 references that are in the map.
- First posted on arXiv 13 May 2025.

## Builds on (most cited first, 14 of 14)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Approximating Likelihood Ratios with Calibrated Discriminative Classifiers](https://mlsanity.com/paper/1506.02169/) (2015)
- [Learning to Pivot with Adversarial Networks](https://mlsanity.com/paper/1611.01046/) (2016)
- [Parameterized Machine Learning for High-Energy Physics](https://mlsanity.com/paper/1601.07913/) (2016)
- [Adversarial Variational Optimization of Non-Differentiable Simulators](https://mlsanity.com/paper/1707.07113/) (2017)
- [Neural Networks for Full Phase-space Reweighting and Parameter Tuning](https://mlsanity.com/paper/1907.08209/) (2019)
- [Positive Difference Distribution for Image Outlier Detection using Normalizing Flows and Contrastive Data](https://mlsanity.com/paper/2208.14024/) (2022)

## Built on by (most cited first, 2 of 2)

- [Contrastive Learning for Interpretable Anomaly Detection at Collider Experiments](https://mlsanity.com/paper/2608.13652/) (2026)
- [Recovering Weak Signals with Normalizing Flows](https://mlsanity.com/paper/2609.06382/) (2026)

## Abstract (arXiv)

Estimating physical parameters from data is a crucial application of machine learning (ML) in the physical sciences. However, systematic uncertainties, such as detector miscalibration, induce data distribution distortions that can erode statistical precision. In both high-energy physics (HEP) and broader ML contexts, achieving uncertainty-aware parameter estimation under these domain shifts remains an open problem. In this work, we address this challenge of uncertainty-aware parameter estimation for a broad set of tasks critical for HEP. We introduce a novel approach based on Contrastive Normalizing Flows (CNFs), which achieves top performance on the HiggsML Uncertainty Challenge dataset. Building on the insight that a binary classifier can approximate the model parameter likelihood ratio, we address the practical limitations of expressivity and the high cost of simulating high-dimensional parameter grids by embedding data and parameters in a learned CNF mapping. This mapping yields a tunable contrastive distribution that enables robust classification under shifted data distributions. Through a combination of theoretical analysis and empirical evaluations, we demonstrate that CNFs, when coupled with a classifier and established frequentist techniques, provide principled parameter estimation and uncertainty quantification through classification that is robust to data distribution distortions.