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Introduction

Autor: José Ángel Hernando Morata

Contributors: Carlos Vázquez Sierra

Institution: Physics Master, Universidade de Santiago de Compostela

email: jose.hernando@usc.es

Versión: February 2023

About these lectures

These lectures are about statistical methods for rare event searches in Particle Physics using Python.

They cover Hypothesis Testing and Confidence Level Intervals. They are based on the excellent lectures on statistics by Prosper (lectures pdf), Cowan (lectures) and Cramer (lectures) given at the CERN Academic Training.

Sometimes, we do an experiment to discover a new particle. If the particle exits in Nature we maybe find only few events. Rare events usually follow poissonian distributions, but statistics are nicely and friendly in the "Gaussian domain".

When we could clain that we have an observation or a discovery of the new particle? And if not, what is the limit in a given observable (i.e. the half lifetime) that we could impose?

In fact, what does it mean discovery, observation, a limit, a confidence level interval? and, how do we compute them from data? These are the questions we try to answer in these lectures.

The beginnig starts with a bifurcation: either we follow a Bayes or a Frequentist path. Be a Bayesian implies do integration (sometime complicated integrals!), and be a frequentist implies either do regression (fits!) or do simulations. But thanks to the current computer power, we can play the frequentist game!

We will use the Python scientic toolkits, Matplotlib, Numpy, Scipy, that are distributed with Anaconda Python.

Table of Contents

Bibliography

[1] "Practical Statistic for LHC physicist," H. B. Prosper, CERN Academic Training Lectures (2015). Lectures PDF

[2] "Statistic for HEP," G. Cowan. CERN Academic Training Lectures (2012). Lectures

[3] "Statistics for Particle Physics," K. Cranmer, CERN Academic Training Lectures (2009). Lectures

[4] "Unified approach to the classical statistical analysis of small signals, "G. J. Feldman and R. D. Cousins, [Phys. Rev. D57 (1998) 3873.])(http://journals.aps.org/prd/abstract/10.1103/PhysRevD.57.3873)

[5] “Asymptotic formulae for likelihood-based tests of new physics,” Glen Cowan, Kyle Cranmer, Eilam Gross, Ofer Vitells. Eur. Phys. J. C71 1554 (2011).

[6] "Incorporating systematic uncertainties into an upper limit," R.D. Cousins and V.L. Highland. Nucl. Instrum. Meth. A320, 331 (1992).

[7] "Confidence Level Computation for Combining Searches with Small Statistics," T. Junk, Nucl. Instrum. Meth. A434, 435 (1999).

[8] "How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics," M. Willians, ArXiv:1006.3019

[9] ROOT, TMVA, RooFit

[10] Anaconda,

[11] "Lectures on Statistics in Theory: Prelude to Statistics in Practice" B. Cousins, ArXiv:1807.05006

[12] "Statistics for physics". D. Tonelli. Invisible School 2019, Canfranc.