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We study the posterior distribution of the Bayesian multiple change-pointregression problem when the number and the locations of the change-points areunknown. 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 exactparametric rate of convergence of the posterior distribution requiresadditional work and assumptions. Additionally, we demonstrate the asymptoticnormality of the segment levels under these assumptions. For inferences on thenumber of change-points, we show that the Bayesian approach can produce aconsistent posterior estimate. Finally, we argue that the point-wise posteriorconvergence property as demonstrated might have bad finite sample performancein that consistent posterior for model selection necessarily implies themaximal squared risk will be asymptotically larger than the optimal$O(1/\sqrt{n})$ rate. This is the Bayesian version of the same phenomenon thathas been noted and studied by other authors.

Posterior Convergence and Model Estimation in Bayesian Change-point Problems