Do pseudomarginal variants actually reduce error relative to plug-in perturbed MCMC?

Determine whether pseudomarginal formulations of perturbed MCMC algorithms achieve strictly smaller sampling error than their direct plug-in counterparts on the original state space, and identify conditions under which any such improvement holds or fails, beyond the possibility that existing theory merely provides more tractable error bounds for pseudomarginal methods.

Background

For many perturbed MCMC algorithms, one can either apply a plug-in estimator directly or employ a pseudomarginal construction on an augmented state space. The latter is typically easier to analyze because its stationary distribution is explicit.

The authors state that it remains largely open whether pseudomarginal variants genuinely have smaller error or simply benefit from more convenient theoretical analyses.