Reweighting of Negative Weights within MC with Uncertainty Quantification

TL;DR

This work addresses the problem of negative weights in MC simulations by deriving a reweighting function $g(oldsymbol{x})=2P_+(oldsymbol{x})-1$ that preserves the original PDF via $PDF_{ ext{reweight}}(oldsymbol{x})=g(oldsymbol{x})igl(PDF_+(oldsymbol{x})+PDF_-(oldsymbol{x})igr)$. When $g$ is known exactly, reweighting reduces variance in MC-based cross sections, and the authors demonstrate this with a double-slit MC example. Since $g(oldsymbol{x})$ is typically unknown, the paper develops a robust uncertainty quantification using a 20-model deep neural network ensemble to model $P_+(oldsymbol{x})$ and propagates uncertainties through event-by-event and PCA-based final-state methods. The PCA-based final observable uncertainty is shown to be robust and computationally efficient, yielding substantial reductions in bin-by-bin MC uncertainties and improved Asimov significance in a Higgs-Z signal region. The methods are demonstrated on ATLAS Open Data Sherpa samples, with broad applicability to any MC sample with negative weights and potential to enhance HL-LHC analyses.

Abstract

High statistical precision is critical for Monte Carlo samples in high energy physics and is degraded by negatively weighted events. This paper investigates a procedure to learn the relationship between the negative and positive weight distributions of any sample, allowing the reduction of statistical uncertainty by reweighting kinematically equivalent events with the same sign. A robust uncertainty quantification method is required for the practical application of such method. Two methods for the estimation of the reweighting uncertainty are developed: one at the event and another at final observable level. The latter method is strongly favored. The gains in statistical precision are then quantified. The method is demonstrated on Sherpa vector boson plus jets samples when using all generated events and when restricted to the signal region of a mock analysis. It is demonstrated to significantly reduce stochastic behavior in sparse MC samples while decreasing the overall uncertainty with a sufficiently well-known reweighting function.

Figures (12)

• Figure 1: Double-slit barrier with slits of width $\delta=0.25$ at positions $\pm\alpha=1$.
• Figure 2: For the left plot, base terms are in blue and the interference terms are in orange. There is a lot of cancellation between the two near a momentum of 0, leading to the much smaller combined term. The right plot shows the reweighting function at each value of $p.$
• Figure 3: The left plot shows the true distribution, the originally sampled distribution, the reweighted distribution, and the positively and negatively weighted events. The right plot has a direct comparison of the true full distribution, the originally weighted distribution, and the reweighted distribution. The top panel compares their histograms, the second panel compares the ratios of the sampled histograms to the true values, the third panel shows the pull on the bin-by-bin MC statistical uncertainties give the uncertainty in each bin and the deviation from the truth, and the bottom panel shows the ratio of the uncertainties of the reweighted bins to the originals. Here $y$ refers to the bin value variable.
• Figure 4: Closure on reweighting of the probability of being positive to the nominal distribution (left), the positive distribution (middle), and the negative distribution (right).
• Figure 5: Histograms of the nominal $P_+$ distribution and the distribution of the positive and negative weights. The left plot has just the nominal distribution, while the right plot adds the reweighted distribution on top. These plots are normalized to the full cross sections of the processes.