Méthodes de Monte Carlo par chaînes de Markov et estimation bayésienne

Abstract

This thesis focuses on Bayesian estimation and the use of Markov Chain Monte Carlo (MCMC) methods. It begins by presenting the theoretical foundations of Bayesian estima- tion, explaining the concepts of prior distribution, posterior distribution, and associated Bayesian estimators. Then, it introduces classical Monte Carlo methods as well as Markov chains, which are fundamental tools for approximating integrals and generating samples from complex distributions. Special attention is given to the Gibbs sampler, a powerful and widely used MCMC algorithm in Bayesian statistics. Finally, practical applications of these methods are presented, particularly in the context of Gaussian models. This work highlights the importance of Bayesian approaches for integrating prior information into statistical analyses and demonstrates the power of MCMC methods in solving inference problems when exact calculations are infeasible.