20231 - BAYESIAN STATISTICAL METHODS

Elementary probability and statistics background is needed.

Bayesian statistical methods have experienced significant advances in the past 20 years, primarily due to their inherent strengths such as logical clarity, flexibility, and the ability to incorporate information from diverse sources. Additionally, these methods have proven capable of tackling complex statistical problems by employing computational techniques based on stochastic simulation methods. Bayesian inference is now widely employed across various scientific disciplines, such as economics, finance, econometrics, marketing, biostatistics, image-processing and more. This course aims to provide an introduction to Bayesian statistics by elucidating its logical principles and underlying concepts. By the end of the course, students will have a solid understanding of Bayesian hierarchical models, linear models, and techniques for model selection. They will also be equipped with the skills to utilize stochastic simulation methods. Special emphasis will be placed on the selection of appropriate statistical models and prior distributions, which serve as the foundation for empirical analysis. Through the use of practical examples and applications utilizing the statistical software R, the course will focus also on developing students' ability to select and implement statistical models that align with the problem at hand.

• Subjective probability: existence, coherence, and properties.
• Bayes' Theorem and statistical Inference as the update of probabilities.
• Predictive approach to statistical inference: exchangeability and de Finetti's representation theorem.
• Selection of prior distributions: conjugate distributions and non-informative priors. 
• Parametric inference: point estimation, interval estimation, hypothesis testing.
• Bayesian hierarchical models, linear and generalized linear models, model selection techniques.
• Stochastic simulation methods: Monte Carlo, Gibbs sampler, and Metropolis-Hastings.
• Bayesian graphical models, Gaussian processes, mixture models.

• Illustrate the concept of subjective probability and its role in Bayesian inference.
• Comprehend the logical foundations of Bayesian analysis.
• Explain the theoretical basis of Bayesian parameter estimation, hypothesis testing, interval estimation, and model selection.
• Identify appropriate prior distributions and the most effective computational techniques based on the statistical problem.

• Select suitable Bayesian statistical models for a given problem.
• Compute posterior distributions using analytical methods and computational techniques.
• Interpret and communicate the results obtained from Bayesian analysis.
• Apply Bayesian statistical methods to real-world problems.
• Critically evaluate the strengths and limitations of Bayesian approaches in different contexts.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)
• Group assignments

The teaching and learning activities are based on lectures that present Bayesian statistics with a main attention to methodology, theory and computational methods. Furthermore, these aspects are illustrated through R code "scripts" available on the Bboard platform, which students can upload to their own computers and use/modify directly to better understand the actual role of various models and proposed initial distributions. The group work, which contributes to the final evaluation, allows students to delve into a topic of their interest (theoretical or applied).

Student evaluations is based on :

• Group assignment, to verify the student's ability to use the methodologies and techniques presented in class in situations other than those explicitly considered in the course.
• Written exam, consisting of exercises and theory questions, which aim to evaluate both the understanding of the proposed methodologies and the student's ability to apply the analytical tools illustrated during the course. 

Grading rule: Let X denote the grade of the written individual exam and let Y be the grade of the group assignment. Then, if Y is greater than or equal to X, the final grade is 0.3*Y+0.7*X. Otherwise, if Y is less than X, the final grade is X.

The course relies, mostly, on the book:

• P.D. HOFF, A first course in Bayesian statistical Methods, New York, Springer-Verlag, 2009.     

Other useful secondary references are listed below:

• A. GELMAN, J.B. CARLIN, H.S. DUNSON, et al., Bayesian Data Analysis, Third Edition, CRC Press, 2013.

Slides and clarification notes summarizing the contents presented in class are also provided. Students who are interested in deepening, individually, specific concepts are provided with additional reading materials upon request. These additional materials are not object of final evaluation.