Gelman & Rubin convergence diagnostic for Ensemble samplers

Question

Background

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.

Question

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.

Answer 1

This post is a bit old but ...

According the the emcee documentation:

"you should not compute the G–R statistic using multiple chains in the same emcee ensemble because the chains are not independent!"

Answer 2

The idea of the Affine Invariant Ensemble Sampler is that each walkers in the ensemble converges to the target density $\pi(\cdot)$. So \begin{align}\Pi(\boldsymbol{\mathrm{X}})&=\Pi(X_1,X_2,X_3,\dots,X_L)\\&=\pi(X_1)\cdot\pi(X_2)\cdot\pi(X_3)\cdots \pi(X_L)\end{align} AIES framework relies on Partial Resampling which is a Generalized Gibbs Step. It ensures that the transition $X_1(t)\to X_1(t+1)$ can depend on $\boldsymbol{\mathrm X}_{[1]}(t)$ and still be able to preserve $\pi$.

However, recently some weakpoints of the Affine Invariant Ensembe Samplers have been show in this article. One of the issues pointed out was that it is tricky to determine if actually all the walkers had burned in even though the trace of the ensemble looks stable already. They used an adapted version of the G-R statistic as well to verify this. This problem gets severe as the parameter space has a dimension higher than 10.