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

Ergodicity of Approximate MCMC Chains with Applications to Large Data Sets