Convergence Diagnostics

A while ago I expressed a desire for better automatic control of Monte Carlo simulations. The magic keyword seems to be "convergence diagnostics".

Statistics practitioners discovered that the Metropolis method (and other Markov Chain methods) were useful for integrals in Bayesian analysis. (handy overview)
I suppose being statistics researchers, they worried more about convergence more that physicists(?) Note there are two convergence worries - has the algorithm reached the long-time limit (stationary distribution), and has it converged to the desired distribution?

The most recent review of the various convergence diagnostics that I've found is by Brooks and Roberts (1999). Not all the diagnostics are applicable to Metropolis sampling (or Metropolis-Hastings, as generalized Metropolis is called).

The other interesting item to look at is "convergence acceleration" and see if any those techniques are useful for physics simulations.