A Bayesian framework for change-point detection with uncertainty quantification

Abstract

We introduce a novel Bayesian method that can detect multiple structural breaks in the mean and variance of a length $T$ time-series. Our method quantifies uncertainty by returning $\alpha$-level credible sets around the estimated locations of the breaks. In the case of a single change in the mean and/or the variance of an independent sub-Gaussian sequence, we prove that our method attains a localization rate that is minimax optimal up to a $\log T$ factor. For an $\alpha$-mixing sequence with dependence, we prove this optimality holds up to $\log^2 T$ factor. For $d$-dimensional mean changes, we show that if $d \gg \log T$ and the mean signal is dense, then our method exactly recovers the location of the change. Our method detects multiple change-points by modularly ``stacking'' single change-point models and searching for a variational approximation to the posterior distribution. This approach is applicable to both continuous and count data. Extensive simulation studies demonstrate that our method is competitive with the state-of-the-art in terms of speed and performance, and we produce credible sets that are an order of magnitude smaller than our competitors without sacrificing nominal coverage guarantees. We apply our method to real data by detecting i) the gating of an ion channel in the outer membrane of a bacterial cell, and ii) changes in the lithological structure of an oil well.