  In this article we perform an asymptotic analysis of Bayesian parallel kernel
density estimators introduced by Neiswanger, Wang and Xing (2014). We derive
the asymptotic expansion of the mean integrated squared error for the full data
posterior estimator and investigate the properties of asymptotically optimal
bandwidth parameters. Our analysis demonstrates that partitioning data into
subsets requires a non-trivial choice of bandwidth parameters that optimizes
the estimation error.
