Data Analysis in High Energy Physics -- Contents -- Preface -- List of Contributors -- 1 Fundamental Concepts -- 1.1 Introduction -- 1.2 Probability Density Functions -- 1.2.1 Expectation Values -- 1.2.2 Moments -- 1.2.3 Associated Functions -- 1.3 Theoretical Distributions -- 1.3.1 The Gaussian Distribution -- 1.3.2 The Poisson Distribution -- 1.3.3 The Binomial Distribution -- 1.3.4 Other Distributions -- 1.4 Probability -- 1.4.1 Mathematical Definition of Probability -- 1.4.2 Classical Definition of Probability -- 1.4.3 Frequentist Definition of Probability -- 1.4.4 Bayesian Definition of Probability -- 1.5 Inference and Measurement -- 1.5.1 Likelihood -- 1.5.2 Frequentist Inference -- 1.5.3 Bayesian Inference -- 1.6 Exercises -- References -- 2 Parameter Estimation -- 2.1 Parameter Estimation in High Energy Physics: Introductory Words -- 2.2 Parameter Estimation: Definition and Properties -- 2.3 The Method of Maximum Likelihood -- 2.3.1 Maximum-Likelihood Solution -- 2.3.2 Properties of the Maximum-Likelihood Estimator -- 2.3.3 Maximum Likelihood and Bayesian Statistics -- 2.3.4 Variance of the Maximum-Likelihood Estimator -- 2.3.5 Minimum-Variance Bound and Experiment Design -- 2.4 The Method of Least Squares -- 2.4.1 Linear Least-Squares Method -- 2.4.2 Non-linear Least-Squares Fits -- 2.5 Maximum-Likelihood Fits:Unbinned, Binned, Standard and Extended Likelihood -- 2.5.1 Unbinned Maximum-Likelihood Fits -- 2.5.2 Extended Maximum Likelihood -- 2.5.3 Binned Maximum-Likelihood Fits -- 2.5.4 Least-Squares Fit to a Histogram -- 2.5.5 Special Topic: Averaging Data with Inconsistencies -- 2.6 Bayesian Parameter Estimation -- 2.7 Exercises -- References -- 3 Hypothesis Testing -- 3.1 Basic Concepts -- 3.1.1 Statistical Hypotheses -- 3.1.2 Test Statistic -- 3.1.3 Critical Region -- 3.1.4 Type I and Type II Errors.

3.1.5 Summary: the Testing Process -- 3.2 Choosing the Test Statistic -- 3.3 Choice of the Critical Region -- 3.4 Determining Test Statistic Distributions -- 3.5 p-Values -- 3.5.1 Significance Levels -- 3.5.2 Inclusion of Systematic Uncertainties -- 3.5.3 Combining Tests -- 3.5.4 Look-Elsewhere Effect -- 3.6 Inversion of Hypothesis Tests -- 3.7 Bayesian Approach to Hypothesis Testing -- 3.8 Goodness-of-Fit Tests -- 3.8.1 Pearson's 2 Test -- 3.8.2 Run Test -- 3.8.3 2 Test with Unbinned Measurements -- 3.8.4 Test Using the Maximum-Likelihood Estimate -- 3.8.5 Kolmogorov-Smirnov Test -- 3.8.6 Smirnov-Cramér-von Mises Test -- 3.8.7 Two-Sample Tests -- 3.9 Conclusion -- 3.10 Exercises -- References -- 4 Interval Estimation -- 4.1 Introduction -- 4.2 Characterisation of Interval Constructions -- 4.3 Frequentist Methods -- 4.3.1 Neyman's Construction -- 4.3.2 Test Inversion -- 4.3.3 Pivoting -- 4.3.4 Asymptotic Approximations -- 4.3.5 Bootstrapping -- 4.3.6 Nuisance Parameters -- 4.4 Bayesian Methods -- 4.4.1 Binomial Efficiencies -- 4.4.2 Poisson Means -- 4.5 Graphical Comparison of Interval Constructions -- 4.6 The Role of Intervals in Search Procedures -- 4.6.1 Coverage -- 4.6.2 Sensitivity -- 4.7 Final Remarks and Recommendations -- 4.8 Exercises -- References -- 5 Classification -- 5.1 Introduction to Multivariate Classification -- 5.2 Classification from a Statistical Perspective -- 5.2.1 Receiver-Operating-Characteristic Curve and the Neyman-Pearson Lemma -- 5.2.2 Supervised Machine Learning -- 5.2.3 Bias-Variance Trade-Off -- 5.2.4 Cross-Validation -- 5.3 Multivariate Classification Techniques -- 5.3.1 Likelihood (Naive Bayes Classifier) -- 5.3.2 k-Nearest Neighbour and Multi-dimensional Likelihood -- 5.3.3 Fisher Linear Discriminant -- 5.3.4 Artificial Neural Networks - Feed-Forward Multi-layer Perceptrons -- 5.3.5 Support Vector Machines.

5.3.6 (Boosted) Decision Trees -- 5.3.7 Boosting and Bagging -- 5.4 General Remarks -- 5.4.1 Pre-processing -- 5.5 Dealing with Systematic Uncertainties -- 5.6 Exercises -- References -- 6 Unfolding -- 6.1 Inverse Problems -- 6.1.1 Direct and Inverse Processes -- 6.1.2 Discretisation and Linear Solution -- 6.1.3 Unfolding Poisson-Distributed Data -- 6.1.4 Convolution and Deconvolution -- 6.1.5 Parametrised Unfolding -- 6.2 Solution with Orthogonalisation -- 6.2.1 Singular Value and Eigenvalue Decomposition -- 6.2.2 Unfolding Using the Least Squares Method -- 6.2.3 Folding Versus Unfolding -- 6.3 Regularisation Methods -- 6.3.1 Norm and Derivative Regularisation -- 6.4 The Discrete Cosine Transformation and Projection Methods -- 6.4.1 Discrete Cosine Transformation -- 6.4.2 Projection Methods -- 6.4.3 Low-Pass Regularisation -- 6.5 Iterative Unfolding -- 6.6 Unfolding Problems in Particle Physics -- 6.6.1 Particle Physics Experiments -- 6.6.2 Unfolding Smooth Distributions -- 6.6.3 Unfolding Non-smooth Distributions -- 6.6.4 Presentation of Regularisation Results -- 6.7 Programs Used for Unfolding in High Energy Physics -- 6.8 Exercise -- References -- 7 Constrained Fits -- 7.1 Introduction -- 7.2 Solution by Elimination -- 7.2.1 Statistical Interpretation -- 7.3 The Method of Lagrange Multipliers -- 7.3.1 Lagrange Multipliers -- 7.3.2 Unmeasured Parameters -- 7.4 The Lagrange Multiplier Problem with Linear Constraintsand Quadratic Objective Function -- 7.4.1 Error Propagation -- 7.4.2 Error Propagation in the Presence of Unmeasured Quantities -- 7.5 Iterative Solution of the Lagrange Multiplier Problem -- 7.5.1 Choosing a Direction -- 7.5.2 Controlling the Step Length -- 7.5.3 Detecting Convergence -- 7.5.4 Finding Initial Values -- 7.5.5 Error Calculation -- 7.6 Further Reading and Web Resources -- 7.7 Exercises -- References.

8 How to Deal with Systematic Uncertainties -- 8.1 Introduction -- 8.2 What Are Systematic Uncertainties? -- 8.3 Detection of Possible Systematic Uncertainties -- 8.3.1 Top-Down Approach -- 8.3.2 Bottom-Up Approach -- 8.3.3 Examples for Detecting Systematics -- 8.4 Estimation of Systematic Uncertainties -- 8.4.1 Some Simple Cases -- 8.4.2 Educated Guesses -- 8.4.3 Cut Variations -- 8.4.4 Combination of Systematic Uncertainties -- 8.5 How to Avoid Systematic Uncertainties -- 8.5.1 Choice of Selection Criteria -- 8.5.2 Avoiding Biases -- 8.5.3 Blind Analyses -- 8.6 Conclusion -- 8.7 Exercise -- References -- 9 Theory Uncertainties -- 9.1 Overview -- 9.2 Factorisation: A Cornerstone of Calculations in QCD -- 9.2.1 The Perturbative Expansion and Uncertainties from Higher Orders -- 9.3 Power Corrections -- 9.3.1 Operator Product Expansion -- 9.3.2 Power Corrections in Cross Sections -- 9.4 The Final State -- 9.4.1 Underlying Event and Multi-Parton Interactions -- 9.4.2 From Partons to Hadrons -- 9.4.3 Monte Carlo Event Generators -- 9.5 From Hadrons to Partons -- 9.5.1 Parametric PDF Uncertainties -- 9.5.2 A Comparison of Recent PDF Sets -- 9.6 Exercises -- References -- 10 Statistical Methods Commonly Used in High Energy Physics -- 10.1 Introduction -- 10.2 Estimating Efficiencies -- 10.2.1 Motivation -- 10.2.2 Trigger Efficiencies and Their Estimates -- 10.2.3 The Counting Method -- 10.2.4 The Tag-and-Probe Method -- 10.2.5 The Bootstrap Method -- 10.2.6 Calculating Uncertainties on Trigger Efficiencies -- 10.3 Estimating the Contributions of Processes to a Dataset:The Matrix Method -- 10.3.1 Estimating the Background Contributions to a Data Sample -- 10.3.2 Extension to Distributions -- 10.3.3 Limitations of the Matrix Method -- 10.4 Estimating Parameters by Comparing Shapes of Distributions:The Template Method -- 10.4.1 Template Shapes.

10.4.2 Including Prior Knowledge -- 10.4.3 Including Efficiencies -- 10.4.4 Including Systematic Uncertainties -- 10.4.5 Systematic Uncertainties Due to the Fitting Procedure -- 10.4.6 Alternative Fitting Methods and Choice of Parameters -- 10.4.7 Extension to Multiple Channels and Multi-Dimensional Templates -- 10.5 Ensemble Tests -- 10.5.1 Generation of Ensembles -- 10.5.2 Results of Ensemble Tests -- 10.6 The Experimenter's Role and Data Blinding -- 10.6.1 The Experimenter's Preconception -- 10.6.2 Variants of Blind Analyses -- 10.7 Exercises -- References -- 11 Analysis Walk-Throughs -- 11.1 Introduction -- 11.2 Search for a Z Boson Decaying into Muons -- 11.2.1 Counting Experiment -- 11.2.2 Profile Likelihood Ratio Analysis -- 11.3 Measurement -- 11.3.1 Introduction -- 11.3.2 Unbinned Likelihood -- 11.3.3 Extracting a Measurement in the Presence of Nuisance Parameters -- 11.3.4 Mass Measurement -- 11.3.5 Testing for Bias and Coverage -- 11.3.6 Systematic Uncertainties -- 11.3.7 Constraints and Combining Measurements -- 11.4 Exercises -- References -- 12 Applications in Astronomy -- 12.1 Introduction -- 12.2 A Survey of Applications -- 12.2.1 The On/Off Problem -- 12.2.2 Image Reconstruction -- 12.2.3 Fitting Cosmological Parameters -- 12.3 Nested Sampling -- 12.4 Outlook and Conclusions -- 12.5 Exercises -- References -- The Authors -- Index.