  When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation
caused by numerical errors is sometimes inevitable. This paper studies how
perturbation of MCMC affects the convergence speed and Monte Carlo estimation
accuracy. Our results show that when the original Markov chain converges to
stationarity fast enough and the perturbed transition kernel is a good
approximation to the original transition kernel, the corresponding perturbed
sampler has similar convergence speed and high approximation accuracy as well.
We discuss two different analysis frameworks: ergodicity and spectral gap, both
are widely used in the literature. Our results can be easily extended to obtain
non-asymptotic error bounds for MCMC estimators.
  We also demonstrate how to apply our convergence and approximation results to
the analysis of specific sampling algorithms, including Random walk Metropolis
and Metropolis adjusted Langevin algorithm with perturbed target densities, and
parallel tempering Monte Carlo with perturbed densities. Finally we present
some simple numerical examples to verify our theoretical claims.
