Simultaneous Credible Regions for Multiple Changepoint Locations

Abstract

Within a Bayesian retrospective framework, we present a way of examining the distribution of changepoints through a novel set estimator. For a given level, α, we aim at smallest sets that cover all changepoints with a probability of at least 1 − α. These so-called smallest simultaneous credible regions, computed for certain values of α, provide parsimonious representations of the possible changepoint locations. In addition, combining them for a range of different α’s enables very informative yet condensed visualizations. Therewith we allow for the evaluation of model choices and the analysis of changepoint data to an unprecedented degree. This approach exhibits superior sensitivity, specificity, and interpretability in comparison with highest density regions, marginal inclusion probabilities, and confidence intervals inferred by stepR. While their direct construction is usually intractable, asymptotically correct solutions can be derived from posterior samples. This leads to a novel NP-complete problem. Through reformulations into an Integer Linear Program we show empirically that a fast greedy heuristic computes virtually exact solutions. Supplementary material for this article is available online.

Keywords:

• Highest density regions
• Integer linear Program
• Model selection
• NP-completeness
• stepR
• Spike and slab

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Correction

Acknowledgments

The authors thank the following persons for their useful hints and corrections: Lisa Koeppel, Paul Fearnhead, Nicolas Wieseke, Johann Jakob Preuß, Lawrence Bardwell, Areesh Mittal, Poppy Miller.

Funding

The first author gratefully acknowledges support from the Landesgraduiertenförderung, Greifswald and the DAAD (ID 57266578).