Communicating Likelihoods with Normalising Flows

TL;DR

This work addresses the challenge of sharing and reusing unbinned likelihoods in high-energy physics by learning a likelihood model (LM) with normalizing flows that map samples to a known base distribution and validating the joint distribution via KS tests. The method includes a standardized training pipeline, a robust loss based on the unbinned likelihood in base space, and a KS-based stopping criterion, implemented in the open-source nabu platform. Through three real-world case studies, the authors demonstrate that the LM can capture complex, non-Gaussian features with strong correlations while achieving compact storage relative to the training data. The results support reliable reinterpretation and downstream inference, and the work outlines clear paths to broader adoption and integration with differentiable workflows and future ML techniques.

Abstract

We present a machine-learning-based workflow to model an unbinned likelihood from its samples. A key advancement over existing approaches is the validation of the learned likelihood using rigorous statistical tests of the joint distribution, such as the Kolmogorov-Smirnov test of the joint distribution. Our method enables the reliable communication of experimental and phenomenological likelihoods for subsequent analyses. We demonstrate its effectiveness through three case studies in high-energy physics. To support broader adoption, we provide an open-source reference implementation, nabu.

Figures (1)

• Figure 1: Summary plots for the three training examples. We show the $p$-value corresponding to the KS and binned $\chi^2$ tests. The "Density" part of each plot shows the 10 bins in $||\vec{\beta}||^2$ and the expected $\chi^2$PDF as a blue curve. Each bin is expected to contain 10% of the testing set for a perfect model. In the "Residuals" part of each plot, the grey, gold, and red coloured bins indicate deviations of less than $1\sigma$, within $[1,2)\sigma$, and more than $2\sigma$, respectively.