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Systematic Uncertainties in Unfolding Considering the Likelihood Formalism

Lydia Brenner, Carsten Burgard, Vincent Alexander Croft

TL;DR

The paper tackles how to treat systematic uncertainties in unfolding by embedding them as nuisance parameters within a likelihood framework, implemented in RooUnfold via a RooFit interface. It introduces an in-likelihood unfolding method that incorporates a Tikhonov regularisation term directly into the likelihood and permits simultaneous estimation of the true distribution and nuisance effects, ensuring proper frequentist coverage. Through toy analytic distributions and a controlled set of nuisance parameters, the study compares the in-likelihood approach to traditional SVD-based methods, showing reduced bias and variance when systematics are included. The approach provides coherent propagation of all uncertainties, extends to higher dimensions or unbinned cases, and delivers practical, coverage-guaranteed regularisation choices for unfolding analyses.

Abstract

This paper describes the treatment of systematic uncertainties in a Likelihood formalism. RooUnfold, which includes most of the unfolding methods that are commonly used in particle physics, is used to compare a newly implemented method inside this toolkit to existing methods. The interface with the RooFit statistical software package is used for the treatment of systematic uncertainties. The RooUnfold package with RooFit interface, commonly called RooFitUnfold, provides a common interface to unfolding algorithms as well as common uniform methods to evaluate their performance in terms of bias, variance and coverage. This paper exploits this common interface to compare the performance of unfolding with a Tikhonov regularisation term directly in the likelihood with an unfolding method that optimises a Tikhonov regularisation applied separately of the likelihood formalism. Comparisons are made with and without the treatment of (systematic) uncertainties and are applied to an example problem.

Systematic Uncertainties in Unfolding Considering the Likelihood Formalism

TL;DR

The paper tackles how to treat systematic uncertainties in unfolding by embedding them as nuisance parameters within a likelihood framework, implemented in RooUnfold via a RooFit interface. It introduces an in-likelihood unfolding method that incorporates a Tikhonov regularisation term directly into the likelihood and permits simultaneous estimation of the true distribution and nuisance effects, ensuring proper frequentist coverage. Through toy analytic distributions and a controlled set of nuisance parameters, the study compares the in-likelihood approach to traditional SVD-based methods, showing reduced bias and variance when systematics are included. The approach provides coherent propagation of all uncertainties, extends to higher dimensions or unbinned cases, and delivers practical, coverage-guaranteed regularisation choices for unfolding analyses.

Abstract

This paper describes the treatment of systematic uncertainties in a Likelihood formalism. RooUnfold, which includes most of the unfolding methods that are commonly used in particle physics, is used to compare a newly implemented method inside this toolkit to existing methods. The interface with the RooFit statistical software package is used for the treatment of systematic uncertainties. The RooUnfold package with RooFit interface, commonly called RooFitUnfold, provides a common interface to unfolding algorithms as well as common uniform methods to evaluate their performance in terms of bias, variance and coverage. This paper exploits this common interface to compare the performance of unfolding with a Tikhonov regularisation term directly in the likelihood with an unfolding method that optimises a Tikhonov regularisation applied separately of the likelihood formalism. Comparisons are made with and without the treatment of (systematic) uncertainties and are applied to an example problem.
Paper Structure (20 sections, 16 equations, 7 figures, 1 table)

This paper contains 20 sections, 16 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definition of the unfolding problem
  3. A likelihood based solution to the unfolding problem
  4. Treatment of systematic uncertainties
  5. Bias, variance and coverage
  6. Studies on example distributions
  7. Distribution definition
  8. Uncertainties considered
  9. Tikhonov regularised methods
  10. Choice of regularisation strength
  11. Results
  12. Conclusion and outlook
  13. Appendix: Nuisance parameters
  14. Overall Scale Factor
  15. Detector Resolution (Smearing)
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Left: The true distribution for the signal plus background model and the corresponding smeared dataset and a simulated measured dataset drawn from the same distribution as the smeared dataset. Right: the response matrix for the signal plus background model, which is populated by the same events as the true distribution shown.
  • Figure 2: Left: Unfolded distribution without systematic uncertainties. Right: Unfolded distribution with systematic uncertainties. For each case the MSE has been minimised, while requiring coverage, to set the appropriate regularisation strength. Due to the fluctuation of the data around the prediction, a perfect match of the unfolded histogram with the Truth distribution is not expected.
  • Figure 3: Left: Unfolded distribution without systematic uncertainties. Right: Unfolded distribution with systematic uncertainties. Using the same regularisation strength as in Figure \ref{['fig:comparison']}. Here a perfect match of the unfolded histogram with the Truth distribution is expected.
  • Figure 4: The systematic variation of scale factors.
  • Figure 5: The systematic variation due to detector smearing.
  • ...and 2 more figures