  A common tool in the practice of Markov Chain Monte Carlo is to use
approximating transition kernels to speed up computation when the desired
kernel is slow to evaluate or intractable. A limited set of quantitative tools
exist to assess the relative accuracy and efficiency of such approximations. We
derive a set of tools for such analysis based on the Hilbert space generated by
the stationary distribution we intend to sample, $L_2(\pi)$. Our results apply
to approximations of reversible chains which are geometrically ergodic, as is
typically the case for applications to Markov Chain Monte Carlo. The focus of
our work is on determining whether the approximating kernel will preserve the
geometric ergodicity of the exact chain, and whether the approximating
stationary distribution will be close to the original stationary distribution.
For reversible chains, our results extend the results of Johndrow et al. [18]
from the uniformly ergodic case to the geometrically ergodic case, under some
additional regularity conditions. We then apply our results to a number of
approximate MCMC algorithms.
