  A current challenge for many Bayesian analyses is determining when to
terminate high-dimensional Markov chain Monte Carlo simulations. To this end,
we propose using an automated sequential stopping procedure that terminates the
simulation when the computational uncertainty is small relative to the
posterior uncertainty. Such a stopping rule has previously been shown to work
well in settings with posteriors of moderate dimension. In this paper, we
illustrate its utility in high-dimensional simulations while overcoming some
current computational issues. Further, we investigate the relationship between
the stopping rule and effective sample size. As examples, we consider two
complex Bayesian analyses on spatially and temporally correlated datasets. The
first involves a dynamic space-time model on weather station data and the
second a spatial variable selection model on fMRI brain imaging data. Our
results show the sequential stopping rule is easy to implement, provides
uncertainty estimates, and performs well in high-dimensional settings.
