• Blended Learning & Flipped Classroom
• Soft Skills

The goal of the course is to enable students to understand and be familiar with the Bayesian approach, its tools and main classes of models. This will enhance students’ abilities to address the statistical analysis of complex systems for which transversal and multidisciplinary skills are required. The Bayesian approach tells us how to update prior beliefs about parameters and hypotheses of a probabilistic model in the light of data, to yield posterior beliefs. In particular, Bayes' Theorem quantifies the problem of how to learn from data.

The course adopts innovative teaching methods (see the corresponding flag):

1. a data analysis project, as a part of the student assessment, to enhance specific soft skills such as Making judgements, Communication skills and Ability in learning (3^rd, 4^th and 5^th elements of the so-called European “Dublin Descriptors” – DD)

2 .  two credits of the full amount of teaching via flipped classroom.

This is a 10 ECTS course, course code # 097659. However, there is a version of it (code # 052502, Bayesian Statistics, 8 ECTS). The only difference between the two courses is that the topics of the 8-credits course do not include neither topic 6 nor models for areal and point-referenced data in topic 7 (see the list of topics below).

On successful completion of the course, students will be able to:

• understand the fundamental principles, tools and models of Bayesian Statistics (DD 1 - knowledge and understanding)
  
• model simple engineering problems in terms of statistical problems using the Bayesian approach to update prior belief in the light of new data (DD 2 - Ability in applying knowledge and understanding)
• select appropriate Bayesian tools to give statistical answers and use appropriate software (R packages, JAGS and BUGS, Stan) for the computation of posterior inference, or to build this software by themselves (DD 2)
• operate and communicate the choices made, making independent judgments, when using Bayesian tools for dealing with statistical problems from the real world (DD 3, 4 and 5)
• work effectively as part of a team

The first part of the course (topic 1) will enable students to understand foundational aspects of Bayesian Statistics (the choice of the prior, for instance), while in the second part (topics 2-7), after the introduction of Markov Chain Monte Carlo techniques for simulation from the posterior, students will be able to use specific statistical models for estimation, prediction or clustering.

1. Basics of Bayesian inference. Likelihood principle, prior and posterior distribution. Bayes' Theorem for dominated models. Posterior summary values. Interpretation of scientific inference via the Bayesian approach. Simple univariate Bayesian models. The three main inferential problem: point estimation, hypothesis testing, interval estimation: comparison between the frequentist and the Bayesian approach. Prior distributions. The choice of a prior distribution, noninformative priors, conjugate priors and their mixtures, semi-conjugate priors. Robustness. Exchangeability and de Finetti's representation theorem for exchangeable sequences. Implications of de Finetti's theorem on the Bayesian approach. Predictive inference. Asymptotic results on the posterior distribution.

2. Simulation methods for Bayesian Statistics. Some results on the theory on general state space Markov chains. Markov chain Monte Carlo methods. Gibbs sampler and Metropolis-Hastings algorithms for computing posterior inference. Implementing Markov chain Monte Carlo: software for Markov chain Monte Carlo (BUGS/JAGS from R), assessing convergence and run-length.

3. Goodness-of-fit and model choice. AIC, BIC, DIC, Predictive Bayesian tail probabilities, log-pseudo marginal likelihood (LPML), WAIC; prior and posterior distributions for the model index, maximum a posteriori (MAP) model selection.

4. Bayesian linear models and generalized linear models. Hierarchical linear models with random effects. Parameter estimation and covariate selection.

5. Bayesian survival analysis / reliability with censored data. Regression models: accelerated failure time and proportional hazards models.

6. Bayesian nonparametric models. The Dirichlet process and generalizations. Bayesian nonparametric mixture models, with application to density estimation and clustering.

7. Introduction to models for longitudinal data, time series, spatial data (point-referenced and areal data).

Part of topics no. 2 and 4 will be given via flipped classrooms (8 h, i.e. 3 classes). For each of these classes, in order to reach the expected learning outcome of each class, students will study the material selected by the instructor and prepare the coding exercises (possible in groups). During the class, some groups will illustrate the exercises in charge to the rest of the students. After the class the instructor will revise the exercises and make this material available to the rest of the class.

The oral exam consists in the presentation of the project, focusing on a statistical problem with data, using Bayesian models, to enhance students’ soft skills such as Making judgements, Communication skills and Ability in learning. As such, 12 h of the innovative teaching will be dedicated to support students in the design of the data analysis project (case study). During these flipped classrooms, the students will apply tools and models from topics 2-7 (see the list above).

The list of topics for the course with code # 052502,  8 ECTS, do not include topic 6 nor models for areal and point-referenced data in topic 7.

Students are required to know the basics of statistical inference and the notion of conditional probability and expectation.

The exam consists of two parts:  a written exam and an oral exam. It is compulsory to register for the written exam, which is scheduled twice in January-February, twice in June-July, and one in September.

The oral exam consists in the illustration of the project analyzing a statistical problem with data, using Bayesian tools and models. As a general rule, the projects will be developed in teams. The evaluation of the projects will be based on the partial presentations given by the teams during the course and a conclusive presentation. The partial presentations contribute to the Innovative Teaching credits (2 credits)

During the conclusive presentation, each students’ team must hand over a project report (10-20 pages).

To obtain a positive final mark, the student should pass with a grade greater or equal than 18/30 both parts of the exam. Final grading: 50% written exam, 50% oral (project) presentation.

After the exam:

• the student will know the fundamental principles, tools and models of Bayesian Statistics
• he/she will be able to apply the acquired knowledge to statistical problems from the real world
• making independent judgments, the student will operate and communicate the choices made, when using Bayesian tools to solve statistical problems from the real world