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 F. Fabbri)
• Module 3, laboratory (lecturer G. Sirri)

For IMAPP students:

• Module 1, theory (lecturer L. Clissa)
• Module 2, exercises and complements (lecturer F. Fabbri)
• Module 3, laboratory (lecturer G. Sirri)
• Module 4, programming (lecturer da F. Betti)
• 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

The lecture slides are made available through the Virtuale e-learning platform. These materials constitute the primary reference for the course and are designed to be complete and sufficient for examination preparation.

For students wishing to gain a deeper understanding of the topics covered in class, clarify specific concepts, or consult alternative presentations of the material, the following reference textbooks are recommended:

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

The course combines lectures, guided problem-solving sessions, and computer laboratory activities to develop both a solid understanding of statistical methods and the ability to apply them to realistic problems in nuclear and subnuclear physics.

Theoretical lectures introduce the main concepts and methodologies, complemented by guided exercises and short interactive quizzes. These quizzes provide immediate feedback on students' understanding, highlight common conceptual difficulties, and allow the instructor to adjust the pace of the course and emphasize topics requiring further clarification.

The laboratory component consists of guided programming and data analysis sessions in which students apply the statistical techniques introduced in class to realistic case studies inspired by nuclear and subnuclear physics analyses. The laboratories make use of software tools and libraries commonly employed in the field, such as ROOT, RooFit, RooStats, and TMVA, with the aim of developing students' ability to implement, validate, and critically interpret modern data analysis workflows.

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 examination consists of a written test divided into three exercises, corresponding to the three modules of the course:

• Module 1: open-ended theoretical questions designed to assess the understanding of the fundamental concepts of probability, statistics, and statistical inference, as well as the ability to critically discuss their properties and fields of application.
• Module 2: a quantitative exercise requiring students to apply the statistical methods introduced during the course to solve a problem, performing the necessary calculations and justifying the adopted approach.
• Module 3: a laboratory-related question in which students are asked to analyse and comment on a short piece of code, explaining its structure, implementation choices, and the underlying statistical model.

The written examination is assessed as a whole, taking into account the correctness of the solutions, the ability to justify the adopted methodology, the appropriate use of scientific terminology, and the clarity of the presentation.

An optional oral examination is also available and is limited to the topics covered in Modules 1 and 2. The oral examination may confirm, increase, or decrease the mark obtained in the written examination. Awarding of the cum laude distinction requires successful completion of the oral examination.

Important: in order to sit the written examination, students must have successfully completed and submitted the practical laboratory assignments associated with Module 3. These assignments are a prerequisite for admission to the examination but do not contribute directly to the final grade.

Use of Generative Artificial Intelligence. The use of Generative Artificial Intelligence (AI) tools (including, but not limited to, ChatGPT, Claude, Gemini, or similar systems) is not permitted during the examination. Any use of such tools during the examination constitutes a violation of the examination regulations and the principles of academic integrity.

Students with temporary or permanent disabilities or Specific Learning Disorders (SLD). Students are advised to contact the University's Disability and Specific Learning Disorders Office well in advance (https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The Office will propose any necessary accommodations, which must be submitted to the course instructor for approval at least 15 days before the examination. The instructor will assess the appropriateness of the requested accommodations in relation to the intended learning outcomes of the course.

Teaching tools

In addition to the lecture slides and teaching materials available through the Virtuale e-learning platform, students are provided with a range of supplementary resources to support learning and independent study.

These include:

• short self-assessment quizzes delivered throughout the course to help students monitor their understanding and identify topics requiring further review;
• multimedia and interactive learning resources (such as concept maps, infographics, interactive web applications, simulations, and other educational materials) designed to highlight the relationships among the main course concepts, provide a broader perspective, and facilitate the understanding of more challenging topics;
• AI-generated audio content intended as an optional study aid. These recordings provide an overview of the course, emphasize the connections between the different topics, and discuss the motivation and applications of the main statistical methods covered during the lectures;
• thematic seminars, delivered by researchers in the field or focused on selected case studies, illustrating the application of modern data analysis techniques to realistic problems in high-energy physics.

These resources are intended to complement, rather than replace, the primary teaching materials of the course.

Office hours

See the website of Luca Clissa

See the website of Federica Fabbri

See the website of Gabriele Sirri