Statistical Methods for Data Analysis in Particle Physics

268 Pages·2017·14.49 MB·English

by  Luca Lista (auth.)

Lecture Notes in Physics 941
Luca Lista
Statistical
Methods for
Data Analysis in
Particle Physics
Second Edition
Lecture Notes in Physics
Volume 941
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Luca Lista
Statistical Methods for Data
Analysis in Particle Physics
Second Edition
123
LucaLista
INFNSezionediNapoli
Napoli,Italy
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-62839-4 ISBN978-3-319-62840-0 (eBook)
DOI10.1007/978-3-319-62840-0
LibraryofCongressControlNumber:2017948232
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Preface
ThisbookstartedasacollectionofmaterialfromacourseoflecturesonStatistical
MethodsforDataAnalysisIgavetoPh.D.studentsinphysicsattheUniversityof
NaplesFedericoIIfrom2009to2017andwassubsequentlyenrichedwithmaterial
fromotherseminarsandlecturesIhavebeeninvitedtogiveinthelastyears.
Theaimofthebookistopresentandelaboratethemainconceptsandtoolsthat
physicistsusetoanalyzeexperimentaldata.
An introductionto probabilitytheory and basic statistics is providedmainly as
refresherlecturesto studentswhodidnottake aformalcourseon statistics before
startingtheirPh.D.ThisalsogivestheopportunitytointroduceBayesianapproach
toprobability,whichisanewtopictomanystudents.
More advanced topics follow, up to recent developmentsin statistical methods
usedforparticlephysics,inparticularfordataanalysesattheLargeHadronCollider.
Manyofthecoveredtoolsandmethodshaveapplicationsinhigh-energyphysics,
buttheirscopecouldwellbeextendedtootherfields.
A shorter version of the course was presented at CERN in November 2009
as lectures on Statistical Methods in LHC Data Analysis for the ATLAS and
CMS experiments. The chapter that discusses discoveries and upper limits was
improvedafter the lectureson the subjectI gavein Autrans,France, atthe IN2P3
School of Statistics in May 2012. I was also invited to conduct a seminar about
Statistical MethodsatGentUniversity,Belgium,in October2014,whichgaveme
theopportunitytoreviewsomeofmymaterialandaddnewexamples.
Note to theSecond Edition
ThesecondeditionofthisbookreflectstheworkIdidinpreparationofthelectures
thatIwasinvitedtogiveduringtheCERN-JINREuropeanSchoolofHigh-Energy
Physics (15–28 June 2016, Skeikampen, Norway). On that occasion, I reviewed,
expanded,andreorderedmymaterial.
v
vi Preface
In addition,with respectto the first edition, I addeda chapter aboutunfolding,
anextendeddiscussionaboutthebestlinearunbiasedestimator,andanintroduction
to machine learning algorithms, in particular artificial neuralnetworks, with hints
aboutdeeplearning,andboosteddecisiontrees.
Acknowledgments
IamgratefultoLouisLyonswhocarefullyandpatientlyreadthefirsteditionofmy
bookand providedusefulcommentsand suggestions.I wouldlike to thankEliam
Grossforprovidingusefulexamplesandforreviewingthesectionsaboutthelook
elsewhere effect. I also received useful comments from Vitaliano Ciulli and from
LuisIsaacRamosGarcia.
IconsideredallfeedbackIreceivedinthepreparationofthissecondedition.
Napoli,Italy LucaLista
Contents
1 ProbabilityTheory ......................................................... 1
1.1 WhyProbabilityMatterstoaPhysicist............................ 1
1.2 TheConceptofProbability......................................... 2
1.3 RepeatableandNon-RepeatableCases ............................ 2
1.4 DifferentApproachestoProbability............................... 3
1.5 ClassicalProbability................................................ 4
1.6 GeneralizationtotheContinuum................................... 6
1.6.1 TheBertrand’sParadox.................................. 7
1.7 AxiomaticProbabilityDefinition .................................. 8
1.8 ProbabilityDistributions............................................ 9
1.9 ConditionalProbability............................................. 9
1.10 IndependentEvents................................................. 10
1.11 LawofTotalProbability............................................ 11
1.12 Average,VarianceandCovariance................................. 12
1.13 TransformationsofVariables....................................... 15
1.14 TheBernoulliProcess .............................................. 16
1.15 TheBinomialProcess............................................... 17
1.16 MultinomialDistribution........................................... 20
1.17 TheLawofLargeNumbers........................................ 21
1.18 FrequentistDefinitionofProbability............................... 22
References.................................................................... 23
2 ProbabilityDistributionFunctions....................................... 25
2.1 Introduction......................................................... 25
2.2 DefinitionofProbabilityDistributionFunction................... 25
2.3 AverageandVarianceintheContinuousCase .................... 27
2.4 Mode,Median,Quantiles........................................... 28
2.5 CumulativeDistribution............................................ 28
2.6 ContinuousTransformationsofVariables ......................... 29
2.7 UniformDistribution ............................................... 30
vii
viii Contents
2.8 GaussianDistribution............................................... 31
2.9 (cid:2)2Distribution...................................................... 32
2.10 LogNormalDistribution ........................................... 33
2.11 ExponentialDistribution............................................ 34
2.12 PoissonDistribution ................................................ 35
2.13 OtherDistributionsUsefulinPhysics ............................. 41
2.13.1 Breit–WignerDistribution............................... 41
2.13.2 RelativisticBreit–WignerDistribution.................. 42
2.13.3 ArgusFunction........................................... 43
2.13.4 CrystalBallFunction.................................... 44
2.13.5 LandauDistribution...................................... 46
2.14 CentralLimitTheorem ............................................. 46
2.15 ProbabilityDistributionFunctionsinMorethanOne
Dimension........................................................... 49
2.15.1 MarginalDistributions................................... 49
2.15.2 IndependentVariables................................... 50
2.15.3 ConditionalDistributions................................ 53
2.16 GaussianDistributionsinTwoorMoreDimensions.............. 54
References.................................................................... 58
3 BayesianApproachtoProbability........................................ 59
3.1 Introduction......................................................... 59
3.2 Bayes’Theorem..................................................... 59
3.3 BayesianProbabilityDefinition.................................... 64
3.4 BayesianProbabilityandLikelihoodFunctions................... 67
3.4.1 RepeatedUseofBayes’TheoremandLearning
Process ................................................... 67
3.5 BayesianInference.................................................. 68
3.5.1 ParametersofInterestandNuisanceParameters....... 69
3.5.2 CredibleIntervals........................................ 70
3.6 BayesFactors ....................................................... 73
3.7 SubjectivenessandPriorChoice................................... 74
3.8 Jeffreys’Prior....................................................... 75
3.9 ReferencePriors .................................................... 76
3.10 ImproperPriors ..................................................... 76
3.11 TransformationsofVariablesandErrorPropagation ............. 79
References.................................................................... 79
4 RandomNumbersandMonteCarloMethods .......................... 81
4.1 PseudorandomNumbers............................................ 81
4.2 PseudorandomGeneratorsProperties.............................. 82
4.3 UniformRandomNumberGenerators............................. 84
4.3.1 RemappingUniformRandomNumbers................ 85
4.4 DiscreteRandomNumberGenerators............................. 85
Contents ix
4.5 NonuniformRandomNumberGenerators......................... 86
4.5.1 NonuniformDistribution from Inversion
oftheCumulativeDistribution.......................... 86
4.5.2 GaussianGeneratorUsingtheCentralLimit
Theorem.................................................. 88
4.5.3 GaussianGeneratorwiththeBox–MullerMethod..... 89
4.6 MonteCarloSampling.............................................. 89
4.6.1 Hit-or-MissMonteCarlo................................ 90
4.6.2 ImportanceSampling.................................... 91
4.7 NumericalIntegrationwithMonteCarloMethods................ 92
4.8 MarkovChainMonteCarlo ........................................ 93
References.................................................................... 95
5 ParameterEstimate ........................................................ 97
5.1 Introduction......................................................... 97
5.2 Inference............................................................. 97
5.3 ParametersofInterest............................................... 98
5.4 NuisanceParameters................................................ 98
5.5 MeasurementsandTheirUncertainties............................ 99
5.5.1 StatisticalandSystematicUncertainties................ 99
5.6 FrequentistvsBayesianInference ................................. 100
5.7 Estimators........................................................... 100
5.8 PropertiesofEstimators............................................ 101
5.8.1 Consistency .............................................. 102
5.8.2 Bias....................................................... 102
5.8.3 MinimumVarianceBoundandEfficiency.............. 102
5.8.4 RobustEstimators........................................ 103
5.9 BinomialDistributionforEfficiencyEstimate .................... 104
5.10 MaximumLikelihoodMethod ..................................... 105
5.10.1 LikelihoodFunction ..................................... 105
5.10.2 ExtendedLikelihoodFunction.......................... 106
5.10.3 GaussianLikelihoodFunctions ......................... 108
5.11 ErrorswiththeMaximumLikelihoodMethod.................... 109
5.11.1 SecondDerivativesMatrix .............................. 109
5.11.2 LikelihoodScan.......................................... 110
5.11.3 PropertiesofMaximumLikelihoodEstimators........ 112
5.12 Minimum(cid:2)2andLeast-SquaresMethods......................... 114
5.12.1 LinearRegression........................................ 115
5.12.2 GoodnessofFitandp-Value............................ 118
5.13 BinnedDataSamples............................................... 118
5.13.1 Minimum(cid:2)2MethodforBinnedHistograms.......... 119
5.13.2 BinnedPoissonianFits .................................. 120