Bayesian uncertainty quantification and structure detection for multiple change points models

Data observed over a long period of time may contain several change points, where the distribution of a variable changes but remains the same over the blocks in between. This useful qualitative structure allows precise estimation and uncertainty quantification for a long vector of parameters. Detecting these change points is another important objective. In this paper, we derive a concentration inequality for an empirical Bayes procedure, obtain the frequentist coverage of a suitable confidence ball of the optimal size constructed from the posterior distribution and study the problem of change point detection. We adopt an oracle approach to quantify the estimation error locally and show that the estimation error of the proposed procedure matches with the oracle rate, thus automatically implying minimax optimality, adaptively over all change point structures. Under a condition on the minimum magnitude of the changes, we show that precisely all change points are detected with high probability, and accompany this with a lower bound result asserting the minimality of that condition. Our results are non-asymptotic and robust in that normality is used only as a working model in the procedure, but the true distribution may not be normal. We discuss important extensions of our results to Hilbert space-valued parameters to address the multiple change point problem for multivariate and functional data. Finally, we describe a possible computational procedure using the simulated annealing method.