Aki Vehtari, Andrew Gelman, Dan Simpson, Bob Carpenter, and Paul-Christian Bürkner have just published a Bayesian Analysis paper about using an improved R factor for MCMC convergence assessment. From the early days of MCMC, convergence assessment has been a recurring (and recurrent!) question in the community. First leading to a flurry of proposals, [which Kerrie, Chantal, and myself reviewwwed in the Valencia 1998 proceedings], and then slowly disintegrating under the onslaughts of reality—i.e. that none could not be 100% foolproof in full generality—…. This included the (possibly now forgotten) single-versus-multiple-chains debate between Charlie Geyer [for single] and Andrew Gelman and Don Rubin [for multiple]. The later introduced an analysis-of-variance R factor, which remains quite popular up to this day, in part for being part of most MCMC software, like BUGS. That this R may fail to identify convergence issues, even in the more recent split version, does not come as a major surprise, since any situation with a long-term influence of the starting distribution may well fail to identify missing (significant) parts of the posterior support. (It is thus somewhat disconcerting to me to see that the main recommendation is to move the bound on R from 1.1 to 1.01, reminding me to some extent of a recent proposal to move the null rejection boundary from 0.05 to 0.005…) Similarly, the ESS may prove a poor signal for convergence or lack thereof, especially because the approximation of the asymptotic variance relies on stationarity assumptions. While multiplying the monitoring tools (as in CODA) helps with identifying convergence issues, looking at a single convergence indicator is somewhat like looking only at a frequentist estimator! (And with greater automation comes greater responsibility—in keeping a critical perspective.)

Looking for a broader perspective, I thus wonder at what we would instead need to assess the lack of convergence of an MCMC chain without much massaging of the said chain. An evaluation of the (Kullback, Wasserstein, or else) distance between the distribution of the chain at iteration n or across iterations, and the true target? A percentage of the mass of the posterior visited so far, which relates to estimating the normalising constant, with a relatively vast array of solutions made available in the recent years? I remain perplexed and frustrated by the fact that, 30 years later, the computed values of the visited likelihoods are not better exploited. Through for instance machine-learning approximations of the target. that could themselves be utilised for approximating the normalising constant and potential divergences from other approximations.