**A**s a sequel to their JRSS B paper, John O’Leary, Guanyang Wang, and [my friend, co-author and former student!] Pierre E. Jacob have recently posted a follow-up paper on maximal coupling for Metropolis-Hastings algorithms, where *maximal* is to be understood in terms of the largest possible probability for the coupled chains to be equal, according to the bound set by the coupling inequality. It made me realise that there is a heap of very recent works in this area.

A question that came up when reading the paper with our PhD students is whether or not the coupled chains stay identical after meeting once. When facing two different targets this seems inevitable and indeed Lemma 2 seems to show that no. A strong lemma that does not [need to] state what happens outside the diagonal Δ.

One of the essential tricks is to optimise several kinds of maximal coupling, incl. one for the Bernoullesque choice of moving, as given on p.3.

Algorithm 1 came as a novelty to me as it first seemed (to me!) the two chains may never meet, but this was before I read the small prints of the transition (proposal) kernel being maximally coupled with itself. While Algorithm 2 may be the earliest example of Metropolis-Hastings coupling I have seen, namely in 1999 in Crete, in connection with a talk by Laird Breyer and Gareth Roberts at a workshop of our ESSS network. As explained by the authors, this solution is not always a maximal coupling for the reason that

min(q¹.q²) min(α¹,α²) ≤ min(q¹α¹,q²α²)

(with q for the transition kernel and α for the acceptance probability). Lemma 1 is interesting in that it describes the probability to un-meet (!) as the surface between one of the move densities and the minimum of the two.

The first solution is to couple by plain Accept-Reject with the first chain being the proposed value and if rejected [i.e. not in C] to generate from the remainder or residual of the second target, in a form of completion of acceptance-rejection (accept when *above* rather than *below*, i.e. in A or A’). This can be shown to be a maximal coupling. Another coupling using reflection residuals works better but requires some spherical structure in the kernel. A further coupling on the acceptance of the Metropolis-Hastings move seems to bring an extra degree of improvement.

In the introduction, the alternatives about the acceptance probability α(·,·), e.g. Metropolis-Hastings versus Barker, are mentioned but would it make a difference to the preferred maximal coupling when using one or the other?

A further comment is that, in larger dimensions, I mean larger than one!, a Gibbsic form of coupling could be considered. In which case it would certainly decrease the coupling probability but may still speed up the overall convergence by coupling more often. See “maximality is sometimes less important than other properties of a coupling, such as the contraction behavior when a meeting does not occur.” (p.8)

As a final pun, I noted that *Vaserstein* is not a typo, as Leonid Vaseršteĭn is a Russian-American mathematician, currently at Penn State.