The Gelman & Rubin (1992) convergence diagnostic requires $M$ MCMC chains to be run, each starting from an over-dispersed estimate of the target population. It then computes a diagnostic based on the between and within-chain variance (this post provides a good summary).
Ensemble MCMC samplers, in particular the Goodman & Weare (2010) Ensemble samplers with affine invariance, use multiple walkers. The proposal distribution for a given walker is then generated (in effect) by the position of all the other walkers.
Ensemble samplers naturally have a number of parallel walkers. Is it (or when is it not) appropriate to use these in place of the $M$ independent chains of the Gelman & Rubin statistic to access convergence?
I have tried this out (see this rough note) this for the emcee implementation of the Goodman & Weare (2010) sampler with some success. However, if I use too many point, the test fails (suggests convergence when it has not converged as seen by eye). I think this is probably due to the fact the samples are not independent, but I'm unsure.
If you have any input on this I'd like to hear.