We consider Metropolis Hastings MCMC in cases where the log of the ratio of target distributions is replaced by an estimator. The estimator is based on m samples from an independent online Monte Carlo simulation. Under some conditions on the distribution of the estimator the process resembles Metropolis Hastings MCMC with a randomized transition kernel. When this is the case there is a correction to the estimated acceptance probability which ensures that the target distribution remains the equilibrium distribution. The simplest versions of the Penalty Method of Ceperley and Dewing (1999), the Universal Algorithm of Ball et al. (2003) and the Single Variable Exchange algorithm of Murray et al. (2006) are special cases. In many applications of interest the correction terms cannot be computed. We consider approximate versions of the algorithms. We show that on average O(m) of the samples realized by a simulation approximating a randomized chain of length n are exactly the same as those of a coupled (exact) randomized chain. Approximation biases Monte Carlo estimates with terms O(1/m) or smaller. This should be compared to the Monte Carlo error which is O(1/sqrt(n)).
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