Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
MONICA MUSIO (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 6 48
[60/65]  MATHEMATICS [65/50 - Ord. 2020]  MATEMATICA PER LA DIDATTICA E LA DIVULGAZIONE 6 48
[60/65]  MATHEMATICS [65/60 - Ord. 2020]  MATEMATICA APPLICATA 6 48

Objectives

KNOWLEDGE AND UNDERSTANDING:
By the end of the course the student knows the basis of Bayesian inference and has the tools for addressing the problems of parametric estimation, predictive inference and hypothesis testing according to the Bayesian viewpoint.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The student is able to face the theoretical problems above and to use statistical software for estimating Bayesian models.
MAKING JUDGMENT:
At the end of the course the student will know how to choose the most effective procedure for solving complex exercises. He will be able to correctly interpret the outputs of a Bayesian analysis.
COMMUNICATION SKILLS:
The student learns how to formalize problems using statistical models and/or is able to understand and use probabilistic modeling.
LEARNING SKILLS:
The student knows how to solve problems dealing with measurement errors and random quantities. Is able to elicit initial laws. He/She is able to collaborate in research centers where statistical data are processed. He/She has the prerequisites to follow advanced inference statistics courses.

Prerequisites

Students must have acquired the knowledge and skills of an introductive course in Probability theory and in Statistics.

Contents

I - Likelihood function (l.f.) 
Notions of statistical experiment, likelihood function, maximum likelihood estimation, likelihood sets; Estimators, sufficient and minimal statistics; Factorization of the l.f.,
Observed and expected information, asymptotic approximations of the l.f.;
l.f. for particular sample models, exponential family.
Subjective nature of probability, historical and critical notes;
Bayes' theorem;
priori and posterior distributions;
Type of stochastic dependence, exchangeability and the predictive approach of probability;
Predictive laws, asymptotic properties of predictive laws, computational aspects.
III - Choice of initial law 
Elicitation of own initial laws, synthesis of experts, conjugated laws;
Principle of insufficient reason, improper laws, reference prior, Jeffreys criterion;
Precautions in choosing initial laws: Cromwell's rule and theorem of precise measurement;
Critical and related aspects to the choice of initial laws.
IV - The decision-making approach 
The decision-making approach from a Bayesian perspective;
Classical problems in Bayesian optics, Bayesian estimates, coverage intervals, asymptotic properties;
Bayesian hypothesis tests, Bayes factors, Jeffreys-Lindley paradox;
Notes on the robust Bayesian analysis for prior and posterior functional;
V - The linear model  (*)
Bayesian setting of the linear model;
The simple, hierarchical regression model, forecast model;
Bayesian discriminant analysis.
VI– Numerical problems  (*)
Laplace integration method, Integration of bayesian function using MC methods.
VII - Elements of stochastic processes  (*)
Markov process;
Chapmann-Kolmogorov implementation,
Poisson and Furry-Yule processes,
Point processes,
Some notes order and extreme statistics
Note 1: compatibly with the available IT tools, some applications will be developed using the environment R
Note 2: the course ends with one of the topics marked with symbol (*)

Teaching Methods

Compatibly with the mixed teaching method foreseen in the Manifesto Accademico 2021-22 as a consequence of the COVID-19 emergency, the tools used for the lectures will be both the blackboard and tablet with projection system via classroom screen and via internet streaming. The course will be integrated with some exercises using the free access software R.

Verification of learning

Compatibly with the indications of the University on how to carry out the exams according to the evolution of the COVID-19 emergency, the exams could be held in the presence or online.
Examination consists of an oral exam and focuses on 3 questions on different topics of the course. Students must be able to perform exercises using the statistical package R.

Texts

Textbook
- Lee P.M. (2012) Bayesian Statistics: An Introduction, 4th Edition, Wiley
- Hoff P. (2009) A First Course in Bayesian Statistical Methods, Springer
- L. Piccinato (1996), Metodi per le decisioni statistiche, Springer-Verlag Italia, Milano.
Other books
- S. J. Press. (2009) Subjective and objective Bayesian Statistics, Wiley, (2nd edn.)
- C.P. Robert. (2007). The bayesian choice, Springer-Verlag, (2nd edn.), N.Y.
- G. Casella, R. L. Berger (2002). Statistical inference, (2nd edn.), Wadsworth Group, CA, USA.
-Eventually notes from the teacher, examples, exercises, as well as programs and procedures developed in R.

To become familiar with the software R:
- S.M. Iacus e G. Masarotto (2007) Laboratorio di statistica con R, MCGraw-Hill
- G.J.Kerns (2011) Introduction to Probability and Statistics Using R (https://cran.r-project.org/web/packages/IPSUR/vignettes/IPSUR.pdf).