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Machine Learning-based Unfolding for Cross Section Measurements in the Presence of Nuisance Parameters

This paper introduces Profile OmniFold, an extension of the machine learning-based OmniFold algorithm that incorporates nuisance parameters to handle uncertainties in detector simulations, demonstrated through both Gaussian examples and a CMS experiment case study.

Original authors: Huanbiao Zhu, Krish Desai, Mikael Kuusela, Vinicius Mikuni, Benjamin Nachman, Larry Wasserman

Published 2026-06-26

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CC BY 4.0

Original authors: Huanbiao Zhu, Krish Desai, Mikael Kuusela, Vinicius Mikuni, Benjamin Nachman, Larry Wasserman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out what a delicious, complex cake tastes like (the true reality), but you can only taste a version of it that has been squished, melted, and slightly altered by a very strange, imperfect oven (the detector).

In the world of particle physics, scientists face this exact problem. They want to know the true properties of tiny particles (like their energy or direction), but their giant detectors (like the CMS experiment at the Large Hadron Collider) distort these measurements. The process of mathematically "un-squishing" the data to find the truth is called unfolding.

For a long time, scientists used a clever machine learning tool called OMNIFOLD to do this. Think of OMNIFOLD as a smart chef who looks at the melted cake and guesses, "Okay, if I add a little bit of this and take away a little bit of that, I can make it look like the original recipe." It works by comparing the real data to a computer simulation of how the oven should work.

The Problem: The Oven Might Be Broken
Here is the catch: The computer simulation isn't perfect. It's based on assumptions about how the oven behaves. Sometimes, the oven is actually slightly different than the simulation says it is—maybe the heat is 10% hotter, or the door seals aren't quite right. In the paper's language, these unknown differences are called nuisance parameters.

If you use the old OMNIFOLD method and assume the oven is perfect when it's actually broken, your guess at the original cake recipe will be wrong. You might think the cake was chocolate when it was actually vanilla, just because your "un-squishing" math was based on the wrong oven settings.

The Solution: Profile OMNIFOLD
The authors of this paper, led by Huanbiao Zhu and colleagues, created a new tool called Profile OMNIFOLD.

Imagine that instead of just guessing the cake recipe, the chef now has a second job: tuning the oven.

The Old Way: The chef guesses the recipe based on a fixed, assumed oven setting. If the setting is wrong, the recipe is wrong.
The New Way (Profile OMNIFOLD): The chef looks at the melted cake and does two things at the same time:
- They adjust the recipe weights (to match the cake).
- They also adjust the oven settings (the nuisance parameters) to see what setting would make the simulation match the real melted cake best.

The algorithm keeps doing this back-and-forth dance: "If I change the oven setting to X, does the recipe look better? If I change the recipe to Y, does the oven setting look better?" It keeps iterating until it finds the perfect combination of True Recipe and Correct Oven Setting.

How They Tested It
The team tested this new method in two ways:

A Simple Math Game: They created a fake scenario using simple bell curves (Gaussian distributions). They knew the "true" answer and the "broken" oven setting. The old method failed to find the true answer, but Profile OMNIFOLD successfully found both the correct shape of the data and the correct oven setting.
Real Physics Data: They used real data from the CMS experiment at CERN, simulating high-energy particle collisions (jets). Again, they introduced a "broken" setting (making the detector resolution 70% worse in the data than in the simulation). The old method produced a biased, incorrect result. Profile OMNIFOLD correctly identified that the detector was "off" and recovered the true particle distribution.

The Bottom Line
The paper claims that by letting the computer learn the "brokenness" of the detector (the nuisance parameters) while it tries to find the true physics, scientists can get much more accurate results. It's like realizing your glasses are smudged and cleaning them while you try to read a book, rather than just squinting and guessing what the words say.

Important Note on Limitations
The authors are careful to point out that this new method isn't magic. It relies on training a second machine learning model to understand how the "brokenness" works. If that training isn't perfect, the final answer might still be a little off. They also note that the method doesn't automatically tell you how "sure" it is about its answer (uncertainty quantification), which is something they plan to work on in the future.

In short: Profile OMNIFOLD is a smarter way to fix distorted data by admitting that our simulations of the machines might be slightly wrong, and fixing those errors at the same time we fix the data.

Technical Summary: Profile OMNIFOLD for Unfolding with Nuisance Parameters

Problem Statement
In particle and nuclear physics, "unfolding" is the statistical inverse problem of recovering the true underlying distribution of a physical quantity (particle-level truth, $x$ ) from observations distorted by detector effects (detector-level data, $y$ ). This relationship is modeled by a Fredholm integral equation of the first kind: $p_Y(y) = \int k(y,x)p_X(x)dx$ , where $k(y,x)$ is the response kernel.

While modern machine learning (ML) methods, particularly the OMNIFOLD algorithm, have successfully addressed high-dimensional unfolding by using classifiers to estimate density ratios within an Expectation-Maximization (EM) framework, they rely on a critical assumption: the detector response kernel used in simulation ( $q(y|x)$ ) perfectly matches the true response ( $p(y|x)$ ). In practice, simulations are only approximately correct and depend on "nuisance parameters" ( $\theta$ ) that encode systematic uncertainties (e.g., jet energy resolution). If $\theta$ is misspecified (i.e., the simulation assumes $\theta=1$ while reality is $\theta \neq 1$ ), standard ML unfolding methods produce biased estimates of the true distribution $p_X$ . Traditional approaches to handle this involve repeating measurements under alternative models, which is computationally expensive and often conservative.

Methodology: Profile OMNIFOLD (POF)
The authors propose Profile OMNIFOLD (POF), an extension of the OMNIFOLD algorithm that jointly estimates the particle-level reweighting function and the nuisance parameters. POF treats the nuisance parameter $\theta$ as an additional unknown in the EM objective function.

The algorithm operates iteratively to maximize the population log-likelihood:
$\ell(\nu, \theta) = \int p(y) \log p(y|\nu, \theta) dy + \log p_0(\theta)$
where $\nu(x)$ is the reweighting function for the Monte Carlo (MC) particle-level density, and $p_0(\theta)$ is a prior (or likelihood term) from auxiliary measurements.

The POF algorithm consists of three steps per iteration $k$ :

Detector-level Reweighting: A classifier estimates the density ratio $r^{(k)}(y) = p(y) / \tilde{q}^{(k)}(y)$ , where $\tilde{q}^{(k)}(y)$ is the MC detector-level density reweighted by both the current particle weights $\nu^{(k)}$ and the current nuisance parameter $\theta^{(k)}$ via a response kernel ratio $W(y,x,\theta) = p(y|x,\theta)/q(y|x)$ .
Particle-level Reweighting: The ratio $r^{(k)}(y)$ is "pulled back" to the particle level to update the weights $\nu^{(k+1)}(x)$ , incorporating the kernel ratio $W$ .
Nuisance Parameter Update: The parameter $\theta$ is updated to $\theta^{(k+1)}$ by maximizing a specific component of the Q-function ( $Q_2$ ) with respect to $\theta$ , holding $\nu^{(k)}$ fixed.

To implement Step 3, the algorithm requires the function $W(y,x,\theta)$ . The paper details a method (based on Chan and Nachman, 2023) to estimate $W$ using two auxiliary classifiers trained on synthetic data generated with varying $\theta$ .

Key Contributions

Algorithmic Extension: The introduction of POF, which integrates nuisance parameter profiling directly into the classifier-based EM unfolding framework. Unlike previous unbinned methods that assume a fixed forward model, POF simultaneously updates the spectrum and the systematic parameters.
Convergence Diagnostics: The authors propose a goodness-of-fit statistic based on the accuracy of the Step 1 classifier. Since a converged solution implies the reweighted MC distribution matches the experimental data, the classifier should be unable to distinguish them (accuracy $\approx 0.5$ ). This statistic is used to select the best solution among multiple runs with different initializations, addressing the non-concavity of the likelihood surface when nuisance parameters are present.
Handling High Dimensions: The method retains the ability of OMNIFOLD to handle high-dimensional data without binning, avoiding the "curse of dimensionality" associated with traditional binned unfolding.

Results
The paper validates POF using two case studies:

Gaussian Example: A controlled 1D and 2D Gaussian simulation where the true nuisance parameter $\theta$ $θ$ (controlling smearing variance) differs from the simulation.
- Findings: Standard OMNIFOLD (assuming $\theta=1$ ) produced significantly biased particle-level distributions. POF successfully recovered the true distribution and estimated $\theta$ close to the ground truth (e.g., $\hat{\theta}=1.48$ vs. true $1.50$). The algorithm converged to the correct solution regardless of initialization when the goodness-of-fit statistic was used for selection.
CMS Open Data Simulation: A realistic simulation of jet transverse momentum spectra from the CMS Experiment at the LHC.
- Setup: The "data" was generated with a jet energy resolution 70% larger than the simulation ( $\theta_{true}=1.7$ vs. $\theta_{sim}=1.0$ ).
- Findings: OMNIFOLD failed to recover the true particle-level spectrum due to the misspecified kernel. POF recovered the true spectrum closely and estimated $\hat{\theta}=1.62$ . The study highlighted that while POF is robust, the choice of initialization matters; poor initializations could lead to local optima, but the proposed goodness-of-fit statistic effectively identified the superior solution.

Significance
The paper argues that Profile OMNIFOLD addresses a fundamental limitation in current machine learning-based unfolding: the assumption of a perfectly specified forward model. By accounting for forward-model uncertainty (nuisance parameters) during the unfolding procedure, POF reduces bias in the recovered particle-level distributions. The authors position this as a practical path for applying ML-based unfolding more robustly in high-energy physics, where detector responses are complex and only known implicitly through simulations. The method preserves the computational efficiency and high-dimensional capabilities of OMNIFOLD while extending its applicability to scenarios with systematic uncertainties.

Limitations and Future Work
The authors note that POF does not have a theoretical convergence guarantee to a global maximum due to the non-concave likelihood surface introduced by nuisance parameters. They emphasize that the estimation of the nuisance parameter is sensitive to the quality of the auxiliary $W$ function estimation. Furthermore, the current implementation does not provide uncertainty quantification for the unfolded distribution or the nuisance parameters, which remains an open problem for future research.

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