Bayesian statistics

Crédit : 6 ECTS
Langue du cours : anglais
Niveau : B2-Avancé ou Indépendant

Volume horaire

• CM : 24 h
• Volume horaire global (hors stage) : 24 h

Compétences à acquérir

Understanding of Bayesian inference and Bayesian decision theory. Understanding and being able to manipulate the asymptotic theory in Bayesian inference : main tools and what they mean.

Description du contenu de l'enseignement

The course will cover different aspects of Bayesian statistics with an emphasis on the theoretical properties of Bayesian methods. The course starts with an introduction Bayesian decision theory from point estimation, to credible regions, testing and model selection and some notion on Bayesian predictive inference. The second part will cover the most important results on Bayesian asymptotics. Part I. Bayesian decision theory : an Introduction

• Prior / Posterior , risks and Bayesian estimators.
• Credible regions.
• Model selection and tests.

Part II: Bayesian asymptotics; in this part, both well and mis-speci?ed models will be considered.

• Asymptotic posterior distribution: in this part we will study asymptotic normality of the posterior, the penalization induced by the prior and the Bernstein von - Mises theorem. Regular and nonregular models will be treated.
• marginal likelihood and consistency of Bayes factors/model selection approaches.
• Empirical Bayes methods. This part will review some results on the asymptotic posterior distribution for parametric empirical Bayes methods.
• Bayesian bootstrap.
• Posterior consistency and posterior convergence rates. This part will ?rst cover the case of statistical loss functions using the theory introduced by L. Schwartz and developed by Ghosal and Van der Vaart

Mode de contrôle des connaissances

examen sur table

Pré-requis recommandés

Probability; basics in statistical inference

Pré-requis obligatoires

probability and basics of statistical inference

Bibliographie, lectures recommandées

• C. P. Robert (2021). The Bayesian Choice.
• S. Ghosal and A. van der Vaart (2017): Fundamentals of Bayesian Nonparametrics.
• A. van der Vaart (1998): Asymptotic Statistics.

Enseignant responsable

JUDITH ROUSSEAU