  Due to the escalating growth of big data sets in recent years, new Bayesian
Markov chain Monte Carlo (MCMC) parallel computing methods have been developed.
These methods partition large data sets by observations into subsets. However,
for Bayesian nested hierarchical models, typically only a few parameters are
common for the full data set, with most parameters being group-specific. Thus,
parallel Bayesian MCMC methods that take into account the structure of the
model and split the full data set by groups rather than by observations are a
more natural approach for analysis. Here, we adapt and extend a recently
introduced two-stage Bayesian hierarchical modeling approach, and we partition
complete data sets by groups. In stage 1, the group-specific parameters are
estimated independently in parallel. The stage 1 posteriors are used as
proposal distributions in stage 2, where the target distribution is the full
model. Using three-level and four-level models, we show in both simulation and
real data studies that results of our method agree closely with the full data
analysis, with greatly increased MCMC efficiency and greatly reduced
computation times. The advantages of our method versus existing parallel MCMC
computing methods are also described.
