- Knowledge and understanding:
the student will understand the Bayesian inferential paradigm and its
difference with respect to classical inferential paradigm.
- Applying knowledge and understanding: the student will be able to specify and estimate a range of models within the
Bayesian approach, assess the quality of the models and interpret the
results. Specifically, the student is required to have a very good use of the Stan software and of the 'rstan' library available in R.
- Making judgements: the student must be able to navigate in the context of the analysis of real data according to a Bayesian approach, with always priority and vigilant attention to the sampling scheme of the data, to their possible hierarchical/multilevel structure and to their granularity.
- Communication skills: the student will be able to effectively communicate the results of data analysis using appropriate tools (including modern techniques for compiling dynamic documents, such as RMarkdown). In addition, the student is also required to make a 'visualization' effort, suitable for the production of graphical tools that summarize complex trends (above all, for example, the use of R libraries such as 'ggplot2').
- Ability to learn: at the end of the course the student will be able to consult theoretical and applied scientific works that use Bayesian statistical techniques, critically analyze the application of the models and algorithms explained in class, and illustrate case studies through the use of probabilistic scientific programming.

Probability calculus, statistical inference

Introduction to Bayesian inference (with refresh of probability calculus
and likelihood inference)
Single parameter models: binomial, normal, Poisson
Bayesian estimate, credibility interval (HPD)
Predictive distribution (PPP), exchangeability
Non informative/ weak informative / reference prior
Multiparameter models: multivariate normal, known and unknown
variance
Asymptotic approximation (parallel with classical inference)
Hierarchical models
Regression model
MCMC general introduction (Gibbs-Metropolis)
Programming an MCMC algorithm in R
Introduction to Stan and use of Stan for estimation.
Variational inference methods in Stan.
Optional: sketch of other approximation methods (Laplace, INLA); model
selection and averaging

Lee - Bayesian Statistics: an introduction, 4th Edition - Wiley
Additional readings:
Gelman, Carlin, Stern, Dunson, Vehtari, Rubin - Bayesian Data Analysis -
CRC press
Albert - Bayesian Computation with R - Springer

The course will be delivered by traditional lectures and practical
computer sessions.
Students will be encouraged to participate at discussion on selected
topics during the lectures.
In the practical sessions the softwares R and STAN will be used to
illustrate some of the main ideas and techniques by analysing some real
datasets

Oral exam and discussion of a practical exercise.
In the oral examination the student will have to prove knowledge of
theoretical results illustrated in the course. The discussion of the practical
exercise will serve the purpose of assessing his capacity of applying the
methods taught 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.