# 41505 - Bayesian Statistical Inference

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

## Course contents

Comparison between classical and Bayesian framework. Open problems in the classical statistical models.

Bayes theorem for events and random variables. Revision of a prior via an experiment.

Bayes inference for events with discrete priors. Odds ratios for couples of events.

Posterior and predictive distributions for a Bernoulli likelihood and discrete and continuous prior distribution. The Beta distribution. Bayesian inference for the normal-normal case with known variance.

Sufficient statistics in Bayesian inference. Natural conjugate distributions.

One parameter exponential family.

Gamma Poisson model.

Gamma and Chi-square distribution.

Inference on the variance of the normal model with known mean.

Inference on the precision of the normal model with known mean.

Two parameters exponential family.

Non informative priors, reference priors.

Alternative formulation for the ignorance on the parameter of the Binomial distribution. Improper priors and Jeffrey's rule. Parameterization of the negative binomial distribution. Exchangeability and hierarchical models. Interval estimation: credibility intervals. Hypothesis testing.

Introduction to a computer code for Bayesian analysis.

Lee P.M. (2012) Bayesian Statistics: An Introduction, 4th Edition, Wiley

Hoff P. (2009) A First Course in Bayesian Statistical Methods, Springer

Further Bibliographical references will be given at the beginning of the course

## Teaching methods

The course is made of lectures and computer sessions

## Assessment methods

The final test (2 hours) will consist of a computer session and a set of questions. The test (2 hours) takes place in one of the computer rooms. The computer session is be mainly devoted to the construction of a model to solve and assess via the software illustrated during the course. The questions focus on the theory and technical points illustrated during the course. After the correction, each candidate will briefly discuss her/his test.

The online test will be performed via “Esami on Line” (EOL). Zoom will be the platform for identification and monitoring.

## Teaching tools

Notes prepared by the teacher will be available.

The course is completed with computer session introducing a software for Bayesian analysis (Stan)

## Office hours

See the website of Daniela Cocchi