What pitfalls should we avoid with Heidelberger-Welch convergence

I'm working through validating a Bayesian mixture model for multi-species occupancy with a collaborator. Initially, we relied on coda::heidel.diag to alert us to convergence failures, and after playing more with fitting our model to toy data we are now more skeptical. We fit the same model to many subsets of the real data, and typically heidel.diag presumes stationarity. What are the pitfalls associated with relying on the Cramer-von-Mises stationarity test, and what additional convergence criteria would be good complements?

We

• have many models to fit, and on fast computers the chains take a couple days to run

• can't investigate all trace plots visually... there are too many

• want something stringent enough to deploy automatically, even if it means more careful attention to specific cases that frequently have, in fact, converged....
• are currently worried about interpreting models that haven't converged
• are working through the recent papers mentioned in this post

Thanks!

• You should not trust one convergence indicator blindly. It only delivers on the explored part of the parameter space. If the sampler has missed a significant part, the indicator will not be able to spot the discrepancy. – Xi'an Apr 28 at 11:41
• @Xi'an, the hope is that we can use one or a few metrics to help us identify which sets of chains are most likely to have failed to converge, not to trust one blindly as a test of true convergence, of course. Given this, is there a pitfall associated with the Heidelberg-Welch diagnostic that we could defend against with another metric? – Michael Apr 28 at 12:58