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Abstract:
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There is a clear 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. Typically, the focus of Bayesian inference is on point and interval estimates for one-dimensional functionals. Approximation methods such as variational Bayes may have poor performance in approximating posterior summaries for such functionals. In small scale problems, Markov chain Monte Carlo (MCMC) remains the gold standard, but such algorithms face major problems in scaling up to big data. We propose a very simple and general Posterior Interval Estimation (PIE) algorithm, which is based on running MCMC in parallel for subsets of the data and averaging quantiles estimated from each subset. We provide strong theoretical guarantees, relate the method to the recent WASP algorithm, and illustrate its empirical performance in several applications.
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