[Paper Review] Practical Statistics for teh LHC

K. Cranmer|arXiv (Cornell University)|Jan 1, 2014

Particle physics theoretical and experimental studies54 references20 citations

TL;DR

This paper provides a comprehensive, pedagogical introduction to statistical methods used in high-energy physics, particularly at the Large Hadron Collider (LHC), emphasizing the construction of statistical models through a 'scientific narrative' framework. It details frequentist and Bayesian procedures, including hypothesis testing, confidence intervals, and likelihood-based inference, with practical applications in signal discovery, exclusion limits, and systematic uncertainty modeling.

ABSTRACT

This document is a pedagogical introduction to statistics for particle physics. Emphasis is placed on the terminology, concepts, and methods being used at the Large Hadron Collider. The document addresses both the statistical tests applied to a model of the data and the modeling itself.

Motivation & Objective

• To provide a clear, accessible introduction to statistical methods used in high-energy physics, especially at the LHC.
• To emphasize the importance of constructing a coherent scientific narrative in statistical modeling as a foundation for robust analysis.
• To formalize and explain key statistical procedures—such as CLs, asymptotic approximations, and importance sampling—used in discovery and exclusion searches.
• To bridge the gap between theoretical statistical concepts and their practical implementation in real LHC analyses.
• To promote consistent, reproducible, and well-justified statistical modeling across experiments using tools like RooStats and HistFactory.

Proposed method

• Uses a narrative-based approach to model data, where the story of signal, backgrounds, and uncertainties is translated into a statistical model.
• Applies both frequentist and Bayesian frameworks, with emphasis on consistent modeling of constraints and systematic uncertainties.
• Employs the likelihood function as the central tool for inference, with explicit treatment of parameter estimation and hypothesis testing.
• Introduces asymptotic formulas using the Asimov dataset to approximate test statistics and expected sensitivity.
• Utilizes importance sampling techniques—naive, phase space slicing, and multiple densities—for efficient Monte Carlo integration.
• Applies the CLs method for exclusion limits and addresses the look-elsewhere effect via trials factor corrections.

Experimental results

Research questions

• RQ1 How can a consistent statistical model be constructed to represent the full scientific narrative of an LHC analysis?
• RQ2 What are the key differences and connections between frequentist and Bayesian approaches in the context of LHC data analysis?
• RQ3 How can systematic uncertainties be reliably modeled and propagated in both frequentist and Bayesian frameworks?
• RQ4 What are the practical implications of the likelihood principle and its violation in frequentist methods?
• RQ5 How can asymptotic approximations and importance sampling improve the efficiency and accuracy of statistical inference in high-dimensional models?

Key findings

• The scientific narrative—describing signal, backgrounds, and uncertainties—forms the essential foundation for building a statistically sound model.
• The Asimov dataset enables accurate approximation of expected test statistics and sensitivity, significantly reducing computational cost.
• Importance sampling techniques, especially phase space slicing and multiple densities, improve the efficiency of Monte Carlo integration in high-dimensional problems.
• The CLs method provides a robust framework for setting upper limits that control the type I error rate in discovery searches.
• Jeffreys’s and reference priors offer objective Bayesian solutions that are invariant under reparameterization, though they can violate the likelihood principle.
• The paper establishes a strong link between statistical procedures and their logical justification, enabling transparent and reproducible analysis in LHC experiments.