Simulation-efficient shortest probability intervals

Abstract

Bayesian highest posterior density (HPD) intervals can be estimated directly from simulations via empirical shortest intervals. Unfortunately, these can be noisy (that is, have a high Monte Carlo error). We derive an optimal weighting strategy using bootstrap and quadratic programming to obtain a more computationally stable HPD, or in general, shortest probability interval (Spin). We prove the consistency of our method. Simulation studies on a range of theoretical and real-data examples, some with symmetric and some with asymmetric posterior densities, show that intervals constructed using Spin have better coverage (relative to the posterior distribution) and lower Monte Carlo error than empirical shortest intervals. We implement the new method in an R package (SPIn) so it can be routinely used in post-processing of Bayesian simulations.

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Acknowledgments

We thank Chia-Hui Huang, Daniel Lee, and Matt Hoffman for research assistance and the National Science Foundation (Grant CNS-1205516), Institute of Education Sciences (Grant DE R305D140059), and Department of Energy (Grant DE-SC0002099) for partial support of this work.

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Correspondence to Andrew Gelman.

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Liu, Y., Gelman, A. & Zheng, T. Simulation-efficient shortest probability intervals. Stat Comput 25, 809–819 (2015). https://doi.org/10.1007/s11222-015-9563-8

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Keywords

  • Bayesian computation
  • Highest posterior density
  • Bootstrap