Contributions computationnelles à la statistique Bayésienne

This thesis presents contributions to the Monte Carlo methodology used in Bayesian statistics. The Bayesian framework is one of the main approaches to statistics and includes a rich methodology to perform inference and model choice. However, as statistical models become more realistic and drift away from the classical assumptions of normality and linearity, computing some of the quantities involved in the statistical analysis becomes a challenge in itself. In particular high-dimensional integrals have to be efficiently approximated, where the integrands can be highly multimodal. Moreover each point-wise evaluation of the integrands can require a lot of computational effort, which results in expensive integration schemes. These integrals are typically approximated using Monte Carlo methods, requiring the ability to sample from general probability distributions. The first chapter of this document explains this motivating context and reviews some of the most generic Monte Carlo techniques. The following chapters aim at improving some of these techniques, at proposing new methods and at analysing their theoretical properties, in the context of sampling from multimodal and computationally expensive probability distributions.