Simultaneous Credible Regions for Multiple Changepoint Locations

We are concerned with quantifying the uncertainty of Bayesian changepoint inference. Therefore, we introduce simultaneous $\alpha$ level credible regions for multiple changepoint locations: these are smallest sets of locations such that with a probability of $1-\alpha$ all changepoints occur inside these sets. We claim that showing such credible regions for a range of different values of $\alpha$ gives a very condensed and informative overview about the possible changepoint locations and the uncertainty of their estimation. Whilst constructing these credible regions is usually intractable, we show how to compute asymptotically correct solutions utilizing a set of samples from the posterior distribution of the changepoints. This leads to a novel NP-complete problem, which we examine in the context of hypergraphs. Furthermore, we reformulate it into an Integer Linear Program (ILP) which allows for the computation of exact solutions. We give a detailed example and compare our credible regions with the highest density regions and the confidence sets for changepoint locations inferred by the R Package stepR. It turns out that our credible regions outperform these methods regarding sensitiveness and specificity.