Description
Book SynopsisThis 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.
Trade Review“The book is important because, as AI and data science continue to shape the future, much interdisciplinary work is being done in many different domains. It is a very good example of interdisciplinary physics research using AI and data science. ... Graduate students are often expected to apply theoretical knowledge. This book will be an invaluable resource for them, to jumpstart their research by getting equipped with the right statistical and data analysis toolsets.” (Gulustan Dogan, Computing Reviews, August 8, 2023)
Table of ContentsPreface to the third edition
Preface to previous edition/s
1 Probability Theory
1.1 Why Probability Matters to a Physicist
1.2 The Concept of Probability
1.3 Repeatable and Non-Repeatable Cases
1.4 Different Approaches to Probability
1.5 Classical Probability
1.6 Generalization to the Continuum
1.7 Axiomatic Probability Definition
1.8 Probability Distributions
1.9 Conditional Probability
1.10 Independent Events
1.11 Law of Total Probability
1.12 Statistical Indicators: Average, Variance and Covariance
1.13 Statistical Indicators for a Finite Sample
1.14 Transformations of Variables
1.15 The Law of Large Numbers
1.16 Frequentist Definition of Probability
References
2 Discrete Probability Distributions
2.1 The Bernoulli Distribution
2.2 The Binomial Distribution
2.3 The Multinomial Distribution
2.4 The Poisson Distribution
References
3 Probability Distribution Functions
3.1 Introduction
3.2 Definition of Probability Distribution Function
3.3 Average and Variance in the Continuous Case
3.4 Mode, Median, Quantiles
3.5 Cumulative Distribution
3.6 Continuous Transformations of Variables
3.7 Uniform Distribution
3.8 Gaussian Distribution
3.9 X^2 Distribution
3.10 Log Normal Distribution
3.11 Exponential Distribution
3.12 Other Distributions Useful in Physics
3.13 Central Limit Theorem
3.14 Probability Distribution Functions in More than One Dimension
3.15 Gaussian Distributions in Two or More Dimensions
References
4 Bayesian Approach to Probability
4.1 Introduction
4.2 Bayes’ Theorem
4.3 Bayesian Probability Definition
4.4 Bayesian Probability and Likelihood Functions
4.5 Bayesian Inference
4.6 Bayes Factors
4.7 Subjectiveness and Prior Choice
4.8 Jeffreys’ Prior
4.9 Reference priors
4.10 Improper Priors
4.11 Transformations of Variables and Error Propagation
References
5 Random Numbers and Monte Carlo Methods
5.1 Pseudorandom Numbers
5.2 Pseudorandom Generators Properties
5.3 Uniform Random Number Generators
5.4 Discrete Random Number Generators
5.5 Nonuniform Random Number Generators
5.6 Monte Carlo Sampling
5.7 Numerical Integration with Monte Carlo Methods
5.8 Markov Chain Monte Carlo
References
6 Parameter Estimate
6.1 Introduction
6.2 Inference
6.3 Parameters of Interest
6.4 Nuisance Parameters
6.5 Measurements and Their Uncertainties
6.6 Frequentist vs Bayesian Inference
6.7 Estimators
6.8 Properties of Estimators
6.9 Binomial Distribution for Efficiency Estimate
6.10 Maximum Likelihood Method
6.11 Errors with the Maximum Likelihood Method
6.12 Minimum X^2 and Least-Squares Methods
6.13 Binned Data Samples
6.14 Error Propagation
6.15 Treatment of Asymmetric Errors
References
7 Combining Measurements
7.1 Introduction
7.2 Simultaneous Fits and Control Regions
7.3 Weighted Average
7.4 X^2 in n Dimensions
7.5 The Best Linear Unbiased Estimator
References
8 Confidence Intervals
8.1 Introduction
8.2 Neyman Confidence Intervals
8.3 Binomial Intervals
8.4 The Flip-Flopping Problem
8.5 The Unified Feldman–Cousins Approach
References
9 Convolution and Unfolding
9.1 Introduction
9.2 Convolution
9.3 Unfolding by Inversion of the Response Matrix
9.4 Bin-by-Bin Correction Factors
9.5 Regularized Unfolding
9.6 Iterative Unfolding
9.7 Other Unfolding Methods
9.8 Software Implementations
9.9 Unfolding in More Dimensions
References
10 Hypothesis Tests
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 a New Signal
References
11 Machine Learning
11.1 Supervised and Unsupervised Learning
11.2 Terminology
11.3 Machine Learning Classification from a Statistical Point of View
11.4 Bias-Variance tradeo
11.5 Overtraining
11.6 Artificial Neural Networks
11.7 Deep Learning
11.8 Convolutional Neural Networks
11.9 Boosted Decision Trees
11.10 Multivariate Analysis Implementations
References
12 Discoveries and Upper Limits
12.1 Searches for New Phenomena: Discovery and Upper Limits
12.2 Claiming a Discovery
12.3 Excluding a Signal Hypothesis
12.4 Combined Measurements and Likelihood Ratio
12.5 Definitions of Upper Limit
12.6 Bayesian Approach
12.7 Frequentist Upper Limits
12.8 Modified Frequentist Approach: the CLs Method
12.9 Presenting Upper Limits: the Brazil Plot
12.10 Nuisance Parameters and Systematic Uncertainties
12.11 Upper Limits Using the Profile Likelihood
12.12 Variations of the Profile-Likelihood Test Statistic
12.13 The Look Elsewhere Effect
References
Index
