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This article presents new methodology for sample-based Bayesian inferencewhen data are partitioned and communication between the parts is expensive, asarises by necessity in the context of "big data" or by choice in order to takeadvantage of computational parallelism. The method, which we call the Laplaceenriched multiple importance estimator, uses new multiple importance samplingtechniques to approximate posterior expectations using samples drawnindependently from the local posterior distributions (those conditioned onisolated parts of the data). We construct Laplace approximations from whichadditional samples can be drawn relatively quickly and improve the methods inhigh-dimensional estimation. The methods are "embarrassingly parallel", make norestriction on the sampling algorithm (including MCMC) to use or choice ofprior distribution, and do not rely on any assumptions about the posterior suchas normality. The performance of the methods is demonstrated and comparedagainst some alternatives in experiments with simulated data.

Importance Sampling Methods for Bayesian Inference with Partitioned Data