# 85238 - BAYESIAN INFERENCE

## Learning outcomes

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. In particular, the student is able to face the theoretical problems above and to use statistical software for estimating Bayesian models.

## 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.

## Teaching tools

The course is completed with computer session introducing a software for Bayesian analysis (Winbugs or Jags)

## Office hours

See the website of Daniela Cocchi