  There is a lack of simple and scalable algorithms for uncertainty
quantification. Bayesian methods quantify uncertainty through posterior and
predictive distributions, but it is difficult to rapidly estimate summaries of
these distributions, such as quantiles and intervals. Variational Bayes
approximations are widely used, but may badly underestimate posterior
covariance. Typically, the focus of Bayesian inference is on point and interval
estimates for one-dimensional functionals of interest. In small scale problems,
Markov chain Monte Carlo algorithms remain the gold standard, but such
algorithms face major problems in scaling up to big data. Various modifications
have been proposed based on parallelization and approximations based on
subsamples, but such approaches are either highly complex or lack theoretical
support and/or good performance outside of narrow settings. We propose a very
simple and general posterior interval estimation algorithm, which is based on
running Markov chain Monte Carlo in parallel for subsets of the data and
averaging quantiles estimated from each subset. We provide strong theoretical
guarantees and illustrate performance in several applications.
