STATISTICS A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2023/2024
Year: 
1
Academic year in which the course will be held: 
2023/2024
Course type: 
Supplementary compulsory subjects
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
80
Detail of lecture’s hours: 
Lesson (48 hours), Exercise (32 hours)
Requirements: 

Essential prerequisite in order to follow the course with profit is the mastery of the topics covered in the course of Probability.

Final Examination: 
Orale

Individual final project in which the tools introduced in the course are used. The focus of the project may be more theoretical (starting from scientific reference articles) or more applied (starting from data of interest that may be provided by the lecturer or sought by the student after discussion with the lecturer).

Assessment: 
Voto Finale

The student will learn the basic concepts of Bayesian statistics, and the related Monte Carlo-type simulation algorithms.

Bayesian approach to inferences is becoming increasingly important in various areas of statistics. The aim of the course is to introduce this approach to parametrical statistical inference problem. The main topics are: introduction to statistical survey methods; the Bayesian paradigm and Bayesian statistical models; methods for assigning a priori distributions; hierarchical models; linear regression model. Monte Carlo and Markov chain Monte Carlo simulation.

Course Main Topics:
• The Bayesian paradigm: priors and posteriors. Some examples
• Prior specification, point estimation, credible intervals, hypothesis testing and model com- parison/selection
• Monte Carlo integration: definition and properties
• Review of Markov chain theory for MCMC purposes
– transition probabilities (for finite and general state spaces) – stationarity
– reversibility
– aperiodicity
– recurrence
– The law of large numbers
– The Central Limit Theorem
– Geometric and uniform ergodicity
• Markov chain Monte Carlo (MCMC) methods
– the Metropolis-Hastings algorithm – the Gibbs sampler
– the Independence sampler
– the Random Walk algorithm
• Burn-in, convergence diagnostics, starting point of the simulation
• The asymptotic variance in MCMC
• Estimating the asymptotic variance in MCMC
• Algorithm performance comparison
• Monte Carlo variance reduction techniques
• Adaptive MCMC algorithms
• Approximate Bayesian Computation

Convenzionale

Lectures and lab sessions with computer. All in the presence of the professor.

- Box, George E. P.; Tiao, George C. “Bayesian inference in statistical analysis.”, Addison-Wesley Series in Behavioral Science: Quantitative Methods. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. xviii+588 pp.
- Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B.
“Bayesian data analysis”. Third edition. Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2014. xiv+661 pp. ISBN: 978-1-4398-4095-5
- Ghosh, Jayanta; Delampady, Mohan; Samanta, Tapas. An introduction to Bayesian analysis. Springer Texts in Statistics. Springer, New York, 2006.
- Didier Dacunha-Castelle and Marie Duflo. Exercices de probabilités et statisti- ques. Tome 1. Collection Mathématiques Appliquées pour la Maitrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1982. Problèmes à temps fixe. [Problems with fixed time].
- Hoff, Peter D., A first course in Bayesian statistical methods. Springer Texts in Statistics. Springer, New York, 2009.
- Mark J. Schervish. Theory of statistics. Springer Series in Statistics. Springer- Verlag, New York, 1995.

Professors

Borrowers