M9211 Bayesian methods

Faculty of Science
Spring 2025
Extent and Intensity
2/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
doc. Mgr. Jan Koláček, Ph.D. (lecturer)
Mgr. Ondřej Pokora, Ph.D. (lecturer)
Guaranteed by
doc. Mgr. Jan Koláček, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
Students are supposed to have theoretical knowledge as well as practical experience in the topics of basic courses of Probability and mathematical statistics and Calculus. The experience in statistical calculations and data analysis in software R is necessary.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This course introduces the basic principles and methods of the Bayesian statistics. The student learns the theoretical principles of these methods and the techniques for calculation of the estimators and their application in inference and prediction. The course also deals with the basics of information theory and with the numerical and simulation methods of the calculations. In the practical classes, the student learns how to calculate the aposterior density and the Bayesian estimators in real problems, how to interpret them and how to compare them with the classical statistical estimators. Further, the student goes through the computer implementation of the methods of numerical integration and Markov-Chain-Monte-Carlo simulations.
Learning outcomes
After completing this course, the student will be able to:
- understand the methods of the Bayesian statistics and to interpret their parameters;
- calculate the Bayesian estimators and to infer in real problems;
- compare the Bayesian and the frequentist approach;
- evaluate the aposterior density by approximations and by the Markov-Chain-Monte-Carlo methods using computers;
- evaluate the I-divergence and the information gained from the experiment.
Syllabus
  • Essentials: conditional probability, Bayes' formula for probabilities.
  • Bayesian statistics: basic idea, comparison with frequentist statistics.
  • Model formulation: sample space, parametric space, prior and posterior distribution.
  • Bayes' theorem: posterior density, formula for discrete and continuous random variables, likelihood function.
  • Crucial properties: likelihood principle, exchangeability, chain rule, sufficient statistics.
  • Bayesian inference: point estimates (posterior mean, MAP), interval estimates (ET interval, HPD region), hypotheses testing.
  • Predictions: predictive distribution.
  • Prior distribution: noninformative, weakly informative, informative, Jeffreys' density, reference, conjugate systems.
  • Models: binomial, Poisson, exponential, gaussian, linear regression model.
  • Hierarchical model: hyperparameters.
  • Simulation methods: Monte Carlo simulations, Monte Carlo integration, Markov Chain Monte Carlo methods, Gibbs' sampler, Metropolis-Hastings algorithm.
  • Introduction to information theory: entropy, mutual information, Kullback-Leibler divergence, informatin gained from experiment.
  • Introduction to statistical decision theory: loss function and estimators.
Literature
  • HOFF, Peter D. A first course in Bayesian statistical methods. Dordrecht: Springer, 2009, ix, 270. ISBN 9780387922997. info
  • ALBERT, Jim. Bayesian computation with R. 2nd ed. Dordrecht: Springer, 2009, xii, 298. ISBN 9780387922973. info
  • GELMAN, Andrew, John B. CARLIN, Hal Steven STERN, David B. DUNSON, Aki VEHTARI and Donald B. RUBIN. Bayesian data analysis. Third edition. Boca Raton: CRC Press/Taylor & Francis, 2014, xiv, 667. ISBN 9781439840955. info
  • CHRISTENSEN, Ronald. Bayesian ideas and data analysis an introduction for scientists and statisticians. Boca Raton: CRC Press, 2011, xvii, 498. ISBN 9781439803547. info
  • ROBERT, Christian. The Bayesian choice : from decision-theoretic foundations to computational implementation. 2nd ed. New York: Springer, 2007, xxiv, 602. ISBN 9780387715988. info
  • PÁZMAN, Andrej. Bayesovská štatistika (Bayesian statistics). Bratislava: Univerzita Komenského Bratislava, 2003, 100 pp. ISBN 80-223-1821-3. info
Teaching methods
Lectures: 2 hours a week. Class exercises: 2 hours a week, work with software R.
Assessment methods
Class exercises: compulsory active participation, solving tasks, homework, ROPOTs, project. The exam takes the form of a short written test and an oral interview (ca. 60 minutes): project defense, theoretical and practical questions, computational procedures. In order to successfully complete the course, it is necessary to fulfill the conditions of the class exercises, submit a project and achieve at least 50 % of the points in the exam.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
https://is.muni.cz/auth/el/sci/jaro2024/M9211/index.qwarp
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents.

The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics.

Assessment in all cases may be in Czech and English, at the student's choice.

The course is also listed under the following terms spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024.
  • Enrolment Statistics (Spring 2025, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2025/M9211