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I received this email last week from Ian Langmore, a postdoc in Columbia:

I’m looking for literature on a subject and can’t find it: I have a Metropolis sampler where the acceptance probability is evaluated with some error. This error is not simply error in evaluation of the target density. It occurs due to the method by which we approximate the acceptance probability.

This is a sensible question, albeit a wee vague… The closest item of work I can think of is the recent paper by Christophe Andrieu and Gareth Roberts, in the Annals of Statistics (2009) following an original proposal by Marc Beaumont. I think there is an early 1990′s paper by Gareth and Jeff Rosenthal where they consider the impact of some approximation effect like real number representation on the convergence but I cannot find it. Of course, the recent particle MCMC JRSS B discussion paper by Christophe, Arnaud Doucet and Roman Hollenstein is a way to bypass the problem. (In a sense ABC is a rudimentary answer as well.) And there must be many other papers on this topic I am not aware of….