Couplings and convergence of Markov Chain Monte Carlo methods

Andreas EBERLE

University of Bonn

Couplings and convergence of Markov Chain Monte Carlo methods

Summary:

In this mini-course we introduce different couplings on continuous state spaces and apply them to quantify contraction and convergence properties of Markov Chain Monte Carlo methods in Wasserstein distances. In the first lecture, we start by introducing several variants of reflection couplings for diffusion processes. These couplings are applied to prove contractivity with explicit rates both for overdamped and for second order Langevin dynamics. In the second lecture, I will explain several ways to carry over the couplings to Markov chains. As a consequence, we derive error bounds for MCMC methods with explicit dependence on the dimension of the state space. Finally, in the last lecture, a related approach will be applied to quantify the convergence to equilibrium for Hamiltonian Monte Carlo.