Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator

When an unbiased estimator of the likelihood is used within an Markov chain Monte Carlo (MCMC) scheme, it is necessary to tradeoff the number of samples used against the computing time. Many samples for the estimator will result in a MCMC scheme which has similar properties to the case where the likelihood is exactly known but will be expensive. Few samples for the construction of the estimator will result in faster estimation but at the expense of slower mixing of the Markov chain. We explore the relationship between the number of samples and the efficiency of the resulting MCMC estimates. Under specific assumptions about the likelihood estimator, we are able to provide guidelines on the number of samples to select for a general Metropolis-Hastings proposal. We provide theory which justifies the use of these assumptions for a large class of models. On a number of examples, we find that the assumptions on the likelihood estimator are accurate.