Stats for HEP

This is a series of walk-throughs of some concepts in statistics relevant for high energy physics (HEP). It is primarily based on (me trying to understand) Refs. [1] and [2], and the CMS combine tool [3]. Both references are useful and highly recommended for reading; the emphasis here is heavily on implementing and visualizing what they discuss in Python - as well as providing some heuristic derivations of some of their formulae - as that is what personally helped me build an intuition for these concepts.

These notes are primarily intended to be followed interactively online as Jupyter notebooks, but are also available as a PDF. Feedback and corrections are very welcome through e-mail or as issues on Github. Finally, for those interested, some more notes and tutorials I have written up, such as on the standard model and machine learning in HEP, are available here.

Table of Contents

• 1. Introduction

I. Frequentist Statistics at the LHC

• 2. Introduction to Likelihoods and Test Statistics
• 3. Hypothesis Testing and Discovery Significance
• 4. Confidence Intervals and Limits
• 5. Expected Significances and Limits

II. Asymptotic Formulae

• 6. Asymptotic Form of the MLE
• 7. Asymptotic Form of the Profile Likelihood Ratio

Coming soon:

7. Asymptotic formulae for signal discovery and limit-setting

Bibliography

[1]

Glen Cowan, Kyle Cranmer, Eilam Gross, and Ofer Vitells. Asymptotic formulae for likelihood-based tests of new physics. Eur. Phys. J. C, 71:1554, 2011. [Erratum: Eur.Phys.J.C 73, 2501 (2013)]. arXiv:1007.1727, doi:10.1140/epjc/s10052-011-1554-0.

[2]

Kyle Cranmer. Practical Statistics for the LHC. In 2011 European School of High-Energy Physics, 267–308. 2014. arXiv:1503.07622, doi:10.5170/CERN-2014-003.267.

[3]

Aram Hayrapetyan and others. The CMS Statistical Analysis and Combination Tool: Combine. Comput. Softw. Big Sci., 8(1):19, 2024. arXiv:2404.06614, doi:10.1007/s41781-024-00121-4.

[4]

Caroline Stiller. Baseline assessment and effect of a supplementary community-based nutrition intervention study on the prevention/treatment of anemia among young Adivasi children in West Bengal, India. PhD thesis, University of Hohenheim, 01 2021.

[5]

ATLAS Collaboration. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B, 716:1–29, 2012. arXiv:1207.7214, doi:10.1016/j.physletb.2012.08.020.