  In many modern applications, difficulty in evaluating the posterior density
makes performing even a single MCMC step slow. This difficulty can be caused by
intractable likelihood functions, but also appears for routine problems with
large data sets. Many researchers have responded by running approximate
versions of MCMC algorithms. In this note, we develop quantitative bounds for
showing the ergodicity of these approximate samplers. We then use these bounds
to study the bias-variance trade-off of approximate MCMC algorithms. We apply
our results to simple versions of recently proposed algorithms, including a
variant of the "austerity" framework of Korratikara et al.
