Posterior convergence and model estimation in Bayesian change-point problems

Abstract

We study the posterior distribution of the Bayesian multiple change-point regression problem when the number and the locations of the change-points are unknown. While it is relatively easy to apply the general theory to obtain the $O(1/\sqrt{n})$ rate up to some logarithmic factor, showing the parametric rate of convergence of the posterior distribution requires additional work and assumptions. Additionally, we demonstrate the asymptotic normality of the segment levels under these assumptions. For inferences on the number of change-points, we show that the Bayesian approach can produce a consistent posterior estimate. Finally, we show that consistent posterior for model selection necessarily implies that the parametric rate for posterior estimation stated previously cannot be uniform over the class of models we consider. This is the Bayesian version of the same phenomenon that has been noted and studied by other authors.