Methods of Bayesian Inference (7 ECTS)

Review of the basic principles of Bayesian Statistics, starting with conditional distributions and Bayes’ theorem: prior and posterior distributions, conjugate families of distributions, and basic methods of Bayesian inference. Introduction to Bayesian hypothesis testing, model comparison and selection, and the Bayes factor. Particular emphasis is placed on multivariate distributions and conjugate models, including the multivariate normal distribution and the Wishart, inverse Wishart, multivariate Student’s t, and Dirichlet distributions, as well as their applications to Bayesian inference for multivariate models. Bayesian inference for normal linear models is then examined, with emphasis on conjugate prior distributions. Finally, Markov chain Monte Carlo (MCMC) methods are introduced, with emphasis on the Metropolis–Hastings and Gibbs algorithms. Applications are presented to (generalized) linear models, multivariate data with missing values, change-point problems, and model-based clustering using finite mixture models.

Recommended Reading

• Παπασταμούλης Π. (2025). Μέθοδοι Μπεϋζιανής Συμπερασματολογίας. 2η εκδοση. Σημειώσεις μαθήματος, ΟΠΑ.
• Robert CP (2007): The Bayesian Choice – From Decision Theoretic Foundations to Computational Implementation. Springer Texts in Statistics.
• Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS. Wiley. Hoboken. USA.
• Carlin B. and Louis T. (2008), Bayes and Empirical Bayes Methods for Data Analysis. 3rd Edition, London: Chapman and Hall.
• Gelman A., Carlin J.B., Stern H.S., Dunson, D.B., Vehtari, A. and Rubin D.B. (2013). Bayesian Data Analysis. Third Edition. Chapman and Hall/CRC.