  Perturbation theory for Markov chains addresses the question how small
differences in the transitions of Markov chains are reflected in differences
between their distributions. We prove powerful and flexible bounds on the
distance of the $n$th step distributions of two Markov chains when one of them
satisfies a Wasserstein ergodicity condition. Our work is motivated by the
recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the
analysis of big data sets. By using an approach based on Lyapunov functions, we
provide estimates for geometrically ergodic Markov chains under weak
assumptions. In an autoregressive model, our bounds cannot be improved in
general. We illustrate our theory by showing quantitative estimates for
approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings
and stochastic Langevin algorithms.
