  Divide-and-conquer based methods for Bayesian inference provide a general
approach for tractable posterior inference when the sample size is large. These
methods divide the data into smaller subsets, sample from the posterior
distribution of parameters in parallel on all the subsets, and combine
posterior samples from all the subsets to approximate the full data posterior
distribution. The smaller size of any subset compared to the full data implies
that posterior sampling on any subset is computationally more efficient than
sampling from the true posterior distribution. Since the combination step takes
negligible time relative to sampling, posterior computations can be scaled to
massive data by dividing the full data into a sufficiently large number of data
subsets. One such approach relies on the geometry of posterior distributions
estimated across different subsets and combines them through their barycenter
in a Wasserstein space of probability measures. We provide theoretical
guarantees on the accuracy of approximation that are valid in many
applications. We show that the geometric method approximates the full data
posterior distribution better than its competitors across diverse simulations
and reproduces known results when applied to a movie ratings database.
