Once learned the fundamental notions of the logic of uncertainty and statistical inference in the previous courses of probability and inference, students must acquire the ability to build statistical models consistent with concrete problems.
For this purpose, the course is intended as both a theoretical and practical introduction to Bayesian inference and statistical modeling.

Knowledge and understanding: Understanding of Bayesian statistical modeling will be developed by relating it to existing knowledge of traditional frequentist approaches. Through the explanation of the philosophical foundations and deviations from conventional frequentist interpretations of probability, the development of Bayesian inference and statistical modeling will be motivated.
Applying Knowledge and understanding: To introduce Bayesian principles in familiar application contexts, we will start with simple binomial and univariate normal models, then move on to hierarchical models and then to simple and multiple regression. Along the way, we will cover different aspects of modeling including model construction, specification of a priori distributions, graphical representations of models, practical aspects of estimation through Monte Carlo methods based on Markov Chain (MCMC), evaluation of the hypotheses and the adaptation of the model to the data, and, finally, the comparison of the models. The examples will be accompanied by demonstrations using three freeware packages, R, Stan and Bugs.
Making judgements: students must show that they know how to choose the most suitable analysis strategy also in the context of analysis of a real data set.
Communication skills: students will be able to effectively communicate the results of data analysis using appropriate tools (including modern techniques for compiling dynamic documents).
Learning skills: students at the end of the course will be able to consult scientific papers, theoretical or applied, that involve the use of Bayesian statistical models.

Prerequisites consist of Probability Calculus for Insurance and Finance and of introductory courses in Inferential Statistics.

1. Introduction: Interpretation and comparison between the classical / frequentist approach and the Bayesian approach to inference
2. Single parameter models: Bernoulli / Binomial models; Normal Models; Other standard distributions
3. Principles of specification of a priori distributions; noninformative and reference, weakly informative, informative priors
4. Bayesian estimation, credible intervals; accumulation of evidence
5. Multiparametric models: Univariate and multivariate Normal models; Multinomial Models
6. Hierarchical Bayesian models: Hierarchical models to integrate information and meta-analysis; exchangeability; complete discussion of a normal hierarchical model
7. Regression models: Bayesian analysis of a linear (ordinary) regression; Hierarchical linear regression models (depending on time constraints)
8. Hints on simulation methods for estimation: direct, numerical and Monte Carlo based on the Markov Chain (MCMC)

- Gelman, Carlin, Stern, Dunson, Vehtari, Rubin (2013) Bayesian Data Analysis (3rd ed.), CRC press
- Ghosh, Delampady, Samanta (2006) An Introduction to Bayesian Analysis - Theory and Methods, Springer
- Gelman, Hill (2007) Data Analysis using Regression and Multilevel/Hierarchical Models, Cambridge University Press
- Albert, Bayesian Computation with R, Springer
- Congdon, P. (2006) Bayesian statistical modelling (2nd ed.), John Wiley & Sons Ltd.
- Congdon, P. (2003) Applied Bayesian modelling, John Wiley & Sons Ltd.
Additional:
- McElreath, R. (2020) Statistical rethinking: A Bayesian course with examples in R and Stan (2nd ed.), CRC Press.
- Gelman, A., Hill, J., & Vehtari, A. (2021) Regression and other stories, Cambridge University Press.
- Kruschke, J. K. (2015) Doing Bayesian data analysis: A tutorial with R and BUGS and Stan (2nd ed.), Academic Press/Elsevier
- Lunn, D., Jackson, C., Best, N., Thomas, A., & Spiegelhalter, D. (2013) The BUGS Book: A Practical Introduction to Bayesian Analysis, Chapman and Hall/CRC.

1. Introduction: Interpretation and comparison between the classical / frequentist approach and the Bayesian approach to inference: probability and different definitions; direct and inverse problems; Bayes' Theorem; brief historical excursus, random uncertainty and epistemic uncertainty; the subjective probability; modern approach to Bayesian statistics
2. Single parameter models: Bernoulli / Binomial models; Normal Models; Other standard distributions
3. Synthesis through a posteriori distribution; Accumulation of evidence
4. Principles of specification of a priori distributions: non-informative and reference, weakly informative and informative priors, conjugation;
5. Multiparametric models: Univariate and multivariate Normal models; Multinomial Models; non-informative and conjugate priors
4. Hierarchical Bayesian models: Hierarchical models to integrate information and meta-analysis; exchangeability; complete discussion of a normal hierarchical model
5. Regression models: Bayesian analysis of a linear (ordinary) regression; Evaluation of hypotheses on parameters and comparison between models; Checking the models; Incorporation of material information; Bayesian updating; Hierarchical linear regression models (depending on time constraint)
6. Notes on simulation methods for estimation: direct, numerical and Monte Carlo based on the Markov Chain (MCMC); Practical introduction to MCMC estimation

Lectures and personal interviews in planned office hours.

At the end of the course, in the exam sessions, the verification will take place with a written test where questions on both theory and applications of the topics of the course can be proposed.
The theory questions will consist of both general questions in which the student must offer a concise, precise and clear treatment of a fundamental topic and more technical and specific questions in which the student must demonstrate an in-depth knowledge of the topics and the ability of using an adequate formal language.
The questions on applications are intended to test the ability to read and interpret the results of an analysis and therefore also demonstrate the practical mastery of the topics covered in the course.

This course covers some topics related to one or more objectives of the 2030 Agenda for the Sustainable Development of United Nations.