Multilevel methods for Gibbs sampling

The multilevel method is a class of algorithms that allowed the sampling probability distribution based on (overdamped) Langevin approximation. The purpose is to sampling an approximation of Bayesian estimator with a controlled cost, in particular with the dimension and the required precision. After a short introduction of the statistical issues we will present multilevel methods for sampling a Gibbs measure and the complexity of these algorithms. The idea of the multilevel is to consider several approximations (level) of the target distribution and do a telescopic sum in order to get the precision of the best approximation for a cost less expensive (than the approximation alone). This estimations are basic Monte-Carlo or a Cesàro averages of Euler schemes of a Langevin diffusion where each level has different time step.