High dimensional Bayesian computation.

Summary Computational Bayesian statistics builds approximations of the a posteriori distribution either by sampling or by building tractable approximations. The contribution of this thesis to the field of Bayesian statistics is the development of new methodology by combining existing methods. Our approaches are better adapted to the dimension or lead to a reduction of the computational cost compared to existing methods.Our first contribution improves the approximate Bayesian computation (ABC) by using the quasi-Monte Carlo (QMC). ABC allows Bayesian inference in models with intractable likelihood. QMC is a variance reduction technique that provides more accurate estimators of integrals. Our second contribution uses QMC for variational inference(VI). VI is a method for constructing tractable approximations to the a posteriori distribution. The third contribution develops an approach to adapt Sequential Monte Carlo (SMC) samplers when using Hamiltonian Monte Carlo (HMC) mutation kernels. SMC samplers allow an unbiased estimation of the model evidence, but they tend to lose performance when the dimension increases. HMC is a Markov chain Monte Carlo technique that has interesting properties when the dimension of the target space increases but is difficult to adapt. By combining the two, we build a sampler that takes advantage of both.