Efficient Estimation of Unfactorizable Systematic Uncertainties

TL;DR

This work addresses the challenge of estimating jointly unfactorizable systematic uncertainties in high-dimensional collider data. It adopts Gaussian process regression with derivative information to model observables as functions of nuisance parameters, and employs a Bayesian Experimental Design strategy to efficiently select training points that minimize predictive uncertainty. Across a simple 1D toy model and 2D/4D high-energy physics efficiency problems, derivative GPs consistently outperform regular GPs and both beat factorization-based baselines, with BED significantly reducing the required number of samples. The approach offers a scalable, nonparametric framework for robust uncertainty quantification in complex experimental settings, improving accuracy and reducing computational burden for precision tests of the Standard Model.

Abstract

Accurate assessment of systematic uncertainties is an increasingly vital task in physics studies, where large, high-dimensional datasets, like those collected at the Large Hadron Collider, hold the key to new discoveries. Common approaches to assessing systematic uncertainties rely on simplifications, such as assuming that the impact of the various sources of uncertainty factorizes. In this paper, we provide realistic example scenarios in which this assumption fails. We introduce an algorithm that uses Gaussian process regression to estimate the impact of systematic uncertainties \textit{without} assuming factorization. The Gaussian process models are enhanced with derivative information, which increases the accuracy of the regression without increasing the number of samples. In addition, we present a novel sampling strategy based on Bayesian experimental design, which is shown to be more efficient than random and grid sampling in our example scenarios.

Figures (20)

• Figure 1: Function ($y$) and and derivative ($\frac{dy}{dx}$) output values of the simple 1D toy model for $x \in [-10, 10]$.
• Figure 2: Panel (a) shows the four training observations (red dots) used to fit the regular GP model. Also shown are the predicted mean $\mu_{\mathrm{test}}$ (blue line) and confidence band (light blue area), together with three functions sampled from the posterior (gray dotted lines). For reference, the true underlying function is depicted in magenta. Similarly, panel (b) shows the predictions of the derivative GP model using the same four observations but augmented by their gradients (red tangent segments).
• Figure 3: First three BED iterations of the regular (a) derivative (b) GP models. The panels show the initial observations (red dots) and the observations selected by the BED strategy (orange dots). Also shown are the predictive mean $\mu_{\mathrm{test}}$ (blue line), the uncertainty band (light blue area), and three functions sampled from the posterior (gray dotted lines). For reference, the true underlying function is depicted in magenta. Along the bottom of each panel, we show the utility function in green. The location of the next observation is indicated by the dotted vertical line, corresponding to the point that maximizes the utility.
• Figure 4: MSE and $\mathrm{Tr}(\Sigma_{\mathrm{test}})$ of the models after each of the 20 BED iterations. The solid lines represent the mean values of 10 calls of the BED strategy with different initial random seeds, and the shaded areas represent the standard deviation.
• Figure 5: Distribution of the $\mathop{\mathrm{p_{\mathrm{T}}}}\limits$ of the two leading jets.