  The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a
perturbed Metropolis-Hastings (MH) algorithm, provides an approach for
approximate sampling when the target distribution is intractable. Assuming the
unperturbed Markov chain is geometrically ergodic, we show explicit estimates
of the difference between the n-th step distributions of the perturbed MCwM and
the unperturbed MH chains. These bounds are based on novel perturbation results
for Markov chains which are of interest beyond the MCwM setting. To apply the
bounds, we need to control the difference between the transition probabilities
of the two chains and to verify stability of the perturbed chain.
