Coupled MCMC with a randomized acceptance probability

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)).

💡 Research Summary

The paper tackles a fundamental difficulty in Metropolis‑Hastings (MH) Monte Carlo: the exact log‑ratio of target densities is often unavailable, and must be estimated from a finite set of samples generated by an independent online simulation. Let Δ denote the true log‑ratio and let Z be a random error obtained from m independent draws; the estimator of the log‑ratio is then Δ̂ = Δ + Z. When the distribution of Z satisfies mild regularity conditions—most importantly symmetry (f_Z(z)=f_Z(−z)) and finite exponential moments—the resulting transition kernel becomes a “randomized” MH kernel. However, simply plugging Δ̂ into the usual acceptance probability α = min{1,exp(Δ̂)} destroys detailed balance, so the target distribution π is no longer invariant.

The authors derive a correction factor c(Z) that restores detailed balance. The corrected acceptance probability is
 α̂(x→y) = min{1,exp(Δ)}·c(Z),
where c(Z) = f_Z(−Z)/f_Z(Z) (or equivalently the ratio of moment‑generating functions E