87944 - Statistical Data Analysis for Nuclear and Subnuclear Physics

Learning outcomes

At the end of the course the student will be acquainted with the main statistical concepts used in physics. After a review of the fundamentals of probability theory, parametric inferential statistics will be introduced, from point estimates and confidence intervals to hypothesis testing and goodness-of-fit. Each item will be addressed both in the Bayesian and frequentist approaches. Dedicated practical sessions will allow the student to become familiar with these conceptual tools by studying applications in nuclear and subnuclear physics.

Course contents

The structure of the course is the following.

For Nuclear and Subnuclear Physics students:

• Module 1, theory (lecturer L. Clissa)
• Module 2, exercises and complements (lecturer M. Negrini)
• Module 3, laboratory (lecturer G. Sirri)

For IMAPP students:

• Module 1, theory (lecturer L. Clissa)
• Module 2, exercises and complements (lecturer M. Negrini)
• Module 3, laboratory (lecturer G. Sirri)
• Module 4, programming (lecturer da F. Giacomini)
• Module 5, computing infrastructure (lecturer da A. Chierici)

Module 1 Program

1. Probability Concepts

   • Definitions: axiomatic, combinatorial, frequentist, and subjectivist
   • Conditional probability
   • Statistical independence
   • Bayes' theorem

2. Random Variables and Distributions

   • Probability density/mass function, cumulative probability function
   • Multivariate distributions
   • Examples of distributions: binomial, multinomial, Poisson, exponential, normal, multivariate normal, chi-squared, Breit-Wigner, Landau
   • Marginal and conditional densities
   • Functions of random variables
   • Characteristic function and distribution moments: expected value, variance, and covariance
   • Central Limit Theorem and Law of Large Numbers
   • Propagation of errors with correlated variables

3. Statistical Inference

   • Fisher information
   • Sample statistics, test statistics, and sufficient statistics
   • Estimators for mean and variance
   • Maximum likelihood method
   • Multi-parameter estimation with uncertainty and correlations
   • Bayesian estimators, Jeffreys priors
   • Least squares method

4. Hypothesis Testing

   • Simple hypotheses
   • Test efficiency and power
   • Neyman-Pearson lemma
   • Linear test, Fisher discriminant
   • Statistical significance, p-values, Look-Elsewhere Effect
   • Chi-squared method for hypothesis testing

5. Confidence Intervals

   • Exact methods: Gaussian and Poisson cases
   • Bayesian method
   • Systematic errors and nuisance parameters
   • Asymptotic properties

6. Multivariate Methods

   • Neural Networks, Boosted Decision Tree

 

Module 2 Program

Exercises and supplements on Monte Carlo methods and unfolding.

 

Module 3 Program

Elements of C++ and ROOT. RooFit workspace, Factory, composite models, multidimensional models. Use of RooStats to calculate confidence intervals, Profile Likelihood, Feldman-Cousins, Bayesian intervals, with and without nuisance parameters. Use of TMVA as a classifier, description of TMVAGui.

Readings/Bibliography

Module 1

• Glen Cowan, Statistical Data Analysis, Oxford Univ. Press, 1998
• (optional, more statistical perspective) Hastie, Trevor, et al. The elements of statistical learning: data mining, inference, and prediction. Vol. 2. New York: springer, 2009.

Modules 2 and 3:

• Glen Cowan, Statistical Data Analysis, Oxford Univ. Press, 1998
• O. Behnke et al., Data Analysis in High Energy Physics: A Practical Guide to Statistical Methods, Wiley, 2013
• A. G. Frodesen, O. Skjeggestad, H. Toft, Probability and Statistics in Particle Physics, Universitetforlaget, 1979
• G. D'Agostini, Bayesian reasoning in data analysis - A critical introduction, World Scientific Publishing, 2003

Teaching methods

Lectures and laboratory sessions using applications for solving practical problems.

Considering the type of activities and teaching methods adopted, attendance in this training activity requires all students attending module 3 to have previously completed modules 1 and 2 on safety training in study places (in e-learning mode).

Assessment methods

The exam consists of a written test composed of 3: 

1. theory questions
2. practical exercise
3. one question for the laboratory part, in which you are asked to comment on a block of code

The oral session is optional, and only regards modules 1 and 2, with the possibility of increaseing/decreasing the written grade by a few points.

To obtain honors, it is necessary to take the oral test.

Important: in order to take the written test, it is necessary to have completed and delivered to the teacher the practical laboratory tests, which do not influence the final grade.

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

Course slides available at Virtuale. Please contact the professors in case of problems.

Office hours

See the website of Luca Clissa

See the website of Matteo Negrini

See the website of Gabriele Sirri