  Inexact Markov Chain Monte Carlo methods rely on Markov chains that do not
exactly preserve the target distribution. Examples include the unadjusted
Langevin algorithm (ULA) and unadjusted Hamiltonian Monte Carlo (uHMC). This
paper establishes bounds on Wasserstein distances between the invariant
probability measures of inexact MCMC methods and their target distributions
with a focus on understanding the precise dependence of this asymptotic bias on
both dimension and discretization step size. Assuming Wasserstein bounds on the
convergence to equilibrium of either the exact or the approximate dynamics, we
show that for both ULA and uHMC, the asymptotic bias depends on key quantities
related to the target distribution or the stationary probability measure of the
scheme. As a corollary, we conclude that for models with a limited amount of
interactions such as mean-field models, finite range graphical models, and
perturbations thereof, the asymptotic bias has a similar dependence on the step
size and the dimension as for product measures.
