  This article presents new methodology for sample-based Bayesian inference
when data are partitioned and communication between the parts is expensive, as
arises by necessity in the context of "big data" or by choice in order to take
advantage of computational parallelism. The method, which we call the Laplace
enriched multiple importance estimator, uses new multiple importance sampling
techniques to approximate posterior expectations using samples drawn
independently from the local posterior distributions (those conditioned on
isolated parts of the data). We construct Laplace approximations from which
additional samples can be drawn relatively quickly and improve the methods in
high-dimensional estimation. The methods are "embarrassingly parallel", make no
restriction on the sampling algorithm (including MCMC) to use or choice of
prior distribution, and do not rely on any assumptions about the posterior such
as normality. The performance of the methods is demonstrated and compared
against some alternatives in experiments with simulated data.
