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

TL;DR

This work shows how to extend machine learning-based unfolding to incorporate nuisance parameters, and presents a new algorithm, called Profile OmniFold, which is demonstrated using a Gaussian example as well as a particle physics case study using simulated data from the CMS Experiment at the Large Hadron Collider.

Abstract

Statistically correcting measured cross sections for detector effects is an important step across many applications. In particle physics, this inverse problem is known as \textit{unfolding}. In cases with complex instruments, the distortions they introduce are often known only implicitly through simulations of the detector. Modern machine learning has enabled efficient simulation-based approaches for unfolding high-dimensional data. Among these, one of the first methods successfully deployed on experimental data is the \textsc{OmniFold} algorithm, a classifier-based Expectation-Maximization procedure. In practice, however, the forward model is only approximately specified, and the corresponding uncertainty is encoded through nuisance parameters. Building on the well-studied \textsc{OmniFold} algorithm, we show how to extend machine learning-based unfolding to incorporate nuisance parameters. Our new algorithm, called Profile \textsc{OmniFold}, is demonstrated using a Gaussian example as well as a particle physics case study using simulated data from the CMS Experiment at the Large Hadron Collider.

Figures (10)

• Figure 1: An overview of the POF algorithm. Portions of the image have been adapted from Andreassen2020 for the original OmniFold algorithm. In step 1, the current particle-level weights $\nu^{(k)}$ are pushed to the detector level with the current nuisance parameters $\theta^{(k)}$, which are used to compute the density ratio $r^{(k)}$. In step 2, the ratio $r^{(k)}$ is pulled pack to the particle level using the same nuisance parameters. In step 3, the nuisance parameters are updated based on the current weights $\nu^{(k)}$ and the density ratio $r^{(k)}$. The procedure is iterated for a fixed number of times.
• Figure 2: Results of unfolding a 2D Gaussian example. Analytic $w$ function is being used in the algorithm. Left: Particle-level kernel density estimates of the truth distribution (black), the MC distribution (blue), and the reweighted MC distributions obtained using the POF (orange) and OF (green) algorithms, each run for 10 iterations. Top-right: Histograms of the four corresponding spectra, aggregated into 50 bins. Bottom-right: The ratio of the truth spectrum to the unfolded spectra.
• Figure 3: Results corresponding to Figure \ref{['fig:Gaussian2DExample_X1']} in detector-level space. Left: Histograms of the corresponding spectra of $Y_1$. Right: Histograms of the corresponding spectra of $Y_2$.
• Figure 4: Evolution of the nuisance parameter and the step-1 classifier’s goodness-of-fit statistic for the results shown in Figure \ref{['fig:Gaussian2DExample_X1']}. Top: Updated estimates $\hat{\theta}$ across iterations for different initializations $\theta^{(0)}$. Bottom: Goodness-of-fit statistic of the step-1 classifier at each iteration.
• Figure 5: Results of unfolding a 2D Gaussian example. Estimated $w$ function is being used in the algorithm. Left: Particle-level kernel density estimates of the truth distribution (black), the MC distribution (blue), and the reweighted MC distributions obtained using the POF (orange) and OF (green) algorithms, each run for 10 iterations. Top-right: Histograms of the four corresponding spectra, aggregated into 50 bins. Bottom-right: The ratio of the truth spectrum to the unfolded spectra.

Theorems & Definitions (8)

• proposition 1
• proposition 2
• Remark 1
• proposition 3
• Remark 2
• proof : Proof of Proposition 1
• proof : Proof of Proposition 2
• proof : Proof of Proposition 3