This third edition expands on the original material. Large portions of the text have been reviewed and clarified. More emphasis is devoted to machine learning including more modern concepts and examples. This book provides the reader with the main concepts and tools needed to perform statistical analyses of experimental data, in particular in the field of high-energy physics (HEP).
It starts with an introduction to probability theory and basic statistics, mainly intended as a refresher from readers’ advanced undergraduate studies, but also to help them clearly distinguish between the Frequentist and Bayesian approaches and interpretations in subsequent applications. Following, the author discusses Monte Carlo methods with emphasis on techniques like Markov Chain Monte Carlo, and the combination of measurements, introducing the best linear unbiased estimator. More advanced concepts and applications are gradually presented, including unfolding and regularization procedures, culminating in the chapter devoted to discoveries and upper limits.
The reader learns through many applications in HEP where the hypothesis testing plays a major role and calculations of look-elsewhere effect are also presented. Many worked-out examples help newcomers to the field and graduate students alike understand the pitfalls involved in applying theoretical concepts to actual data.
Author(s): Luca Lista
Series: Lecture Notes in Physics, 1010
Edition: 3
Publisher: Springer
Year: 2023
Language: English
Pages: 357
City: Cham
Preface
Contents
List of Figures
List of Tables
List of Examples
1 Introduction to Probability and Inference
1.1 Why Probability Matters to a Physicist
1.2 Random Processes and Probability
1.3 Different Approaches to Probability
1.4 Classical Probability
1.5 Problems with the Generalization to the Continuum
1.6 The Bertrand's Paradox
1.7 Axiomatic Probability Definition
1.8 Conditional Probability
1.9 Independent Events
1.10 Law of Total Probability
1.11 Inference
1.12 Measurements and Their Uncertainties
1.13 Statistical and Systematic Uncertainties
1.14 Frequentist vs Bayesian Inference
References
2 Discrete Probability Distributions
2.1 Introduction
2.2 Joint and Marginal Probability Distributions
2.3 Conditional Distributions and Chain Rule
2.4 Independent Random Variables
2.5 Statistical Indicators: Average, Variance, and Covariance
2.6 Statistical Indicators for Finite Samples
2.7 Transformations of Variables
2.8 The Bernoulli Distribution
2.9 The Binomial Distribution
2.10 The Multinomial Distribution
2.11 The Poisson Distribution
2.12 The Law of Large Numbers
2.13 Law of Large Numbers and Frequentist Probability
References
3 Probability Density Functions
3.1 Introduction
3.2 Definition of Probability Density Function
3.3 Statistical Indicators in the Continuous Case
3.4 Cumulative Distribution
3.5 Continuous Transformations of Variables
3.6 Marginal Distributions
3.7 Uniform Distribution
3.8 Gaussian Distribution
3.9 χ2 Distribution
3.10 Log Normal Distribution
3.11 Exponential Distribution
3.12 Gamma Distribution
3.13 Beta Distribution
3.14 Breit–Wigner Distribution
3.15 Relativistic Breit–Wigner Distribution
3.16 Argus Distribution
3.17 Crystal Ball Function
3.18 Landau Distribution
3.19 Mixture of PDFs
3.20 Central Limit Theorem
3.21 Probability Distributions in Multiple Dimension
3.22 Independent Variables
3.23 Covariance, Correlation, and Independence
3.24 Conditional Distributions
3.25 Gaussian Distributions in Two or More Dimensions
References
4 Random Numbers and Monte Carlo Methods
4.1 Pseudorandom Numbers
4.2 Properties of Pseudorandom Generators
4.3 Uniform Random Number Generators
4.4 Inversion of the Cumulative Distribution
4.5 Random Numbers Following a Finite Discrete Distribution
4.6 Gaussian Generator Using the Central Limit Theorem
4.7 Gaussian Generator with the Box–Muller Method
4.8 Hit-or-Miss Monte Carlo
4.9 Importance Sampling
4.10 Numerical Integration with Monte Carlo Methods
4.11 Markov Chain Monte Carlo
References
5 Bayesian Probability and Inference
5.1 Introduction
5.2 Bayes' Theorem
5.3 Bayesian Probability Definition
5.4 Decomposing the Denominator in Bayes' Formula
5.5 Bayesian Probability Density and Likelihood Functions
5.6 Bayesian Inference
5.7 Repeated Observations of a Gaussian Variable
5.8 Bayesian Inference as Learning Process
5.9 Parameters of Interest and Nuisance Parameters
5.10 Credible Intervals
5.11 Bayes Factors
5.12 Subjectiveness and Prior Choice
5.13 Jeffreys' Prior
5.14 Reference Priors
5.15 Improper Priors
5.16 Transformations of Variables and Error Propagation
References
6 Frequentist Probability and Inference
6.1 Frequentist Definition of Probability
6.2 Estimators
6.3 Properties of Estimators
6.4 Robust Estimators
6.5 Maximum Likelihood Method
6.6 Likelihood Function
6.7 Binned and Unbinned Fits
6.8 Numerical Implementations
6.9 Likelihood Function for Gaussian Distribution
6.10 Errors of Maximum Likelihood Estimates
6.11 Covariance Matrix Estimate
6.12 Likelihood Scan
6.13 Properties of Maximum Likelihood Estimators
6.14 Extended Likelihood Function
6.15 Minimum χ2 and Least Squares Methods
6.16 Linear Regression
6.17 Goodness of Fit and p-Value
6.18 Minimum χ2 Method for Binned Histograms
6.19 Binned Poissonian Fits
6.20 Constrained Fitting
6.21 Error Propagation
6.22 Simple Cases of Error Propagation
6.23 Propagation of Asymmetric Errors
6.24 Asymmetric Error Combination with a Linear Model
References
7 Combining Measurements
7.1 Introduction
7.2 Control Regions
7.3 Simultaneous Fits
7.4 Weighted Average
7.5 Combinations with χ2 in n Dimensions
7.6 The Best Linear Unbiased Estimator (BLUE)
7.7 Quantifying the Importance of Individual Measurements
7.8 Negative Weights
7.9 Iterative Application of the BLUE Method
References
8 Confidence Intervals
8.1 Introduction
8.2 Neyman Confidence Intervals
8.3 Construction of the Confidence Belt
8.4 Inversion of the Confidence Belt
8.5 Binomial Intervals
8.6 The Flip-Flopping Problem
8.7 The Unified Feldman–Cousins Approach
References
9 Convolution and Unfolding
9.1 Introduction
9.2 Convolution
9.3 Convolution and Fourier Transform
9.4 Discrete Convolution and Response Matrix
9.5 Effect of Efficiency and Background
9.6 Unfolding by Inversion of the Response Matrix
9.7 Bin-by-Bin Correction Factors
9.8 Regularized Unfolding
9.9 Tikhonov Regularization
9.10 L-Curve Scan
9.11 Iterative Unfolding
9.12 Adding Background in Iterative Unfolding
9.13 Other Unfolding Methods
9.14 Software Implementations
9.15 Unfolding in More Dimensions
References
10 Hypothesis Testing
10.1 Introduction
10.2 Test Statistic
10.3 Type I and Type II Errors
10.4 Fisher's Linear Discriminant
10.5 The Neyman–Pearson Lemma
10.6 Projective Likelihood Ratio Discriminant
10.7 Kolmogorov–Smirnov Test
10.8 Wilks' Theorem
10.9 Likelihood Ratio in the Search for New Signals
References
11 Machine Learning
11.1 Supervised and Unsupervised Learning
11.2 Machine-Learning Classification from a Statistics Perspective
11.3 Terminology
11.4 Curve Fit as Machine-Learning Problem
11.5 Undertraining and Overtraining
11.6 Bias-Variance Trade-Off
11.7 Logistic Regression
11.8 Softmax Regression
11.9 Support Vector Machines
11.10 Artificial Neural Networks
11.11 Deep Learning
11.12 Deep Learning in Particle Physics
11.13 Convolutional Neural Networks
11.14 Recursive Neural Networks
11.15 Graph Neural Networks
11.16 Random Forest and Boosted Decision Trees
11.17 Unsupervised Learning
11.18 Clustering
11.19 Anomaly Detection
11.20 Autoencoders
11.21 Reinforcement Learning
11.22 Generative Adversarial Network
11.23 No Free Lunch Theorem
References
12 Discoveries and Limits
12.1 Searches for New Phenomena: Discovery and Limits
12.2 p-Values and Discoveries
12.3 Significance Level
12.4 Signal Significance and Discovery
12.5 Significance for Poissonian Counting Experiments
12.6 Significance with Likelihood Ratio
12.7 Significance Evaluation with Toy Monte Carlo
12.8 Excluding a Signal Hypothesis
12.9 Combined Measurements and Likelihood Ratio
12.10 Definitions of Upper Limit
12.11 Bayesian Approach
12.12 Bayesian Upper Limits for Poissonian Counting
12.13 Limitations of the Bayesian Approach
12.14 Frequentist Upper Limits
12.15 Frequentist Upper Limits for Counting Experiments
12.16 Frequentist Limits in Case of Discrete Variables
12.17 Feldman–Cousins Unified Approach
12.18 Modified Frequentist Approach: The CLs Method
12.19 Presenting Upper Limits: The Brazil Plot
12.20 Nuisance Parameters and Systematic Uncertainties
12.21 Nuisance Parameters with the Bayesian Approach
12.22 Hybrid Treatment of Nuisance Parameters
12.23 Event Counting Uncertainties
12.24 Upper Limits Using the Profile Likelihood
12.25 Variations of the Profile Likelihood Test Statistic
12.26 Test Statistic for Positive Signal Strength
12.27 Test Statistic for Discovery
12.28 Test Statistic for Upper Limits
12.29 Higgs Test Statistic
12.30 Asymptotic Approximations
12.31 Asimov Data Sets
12.32 The Look Elsewhere Effect
12.33 Trial Factors
12.34 Look Elsewhere Effect in More Dimensions
References
Index
