  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 exact
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 argue that the point-wise posterior
convergence property as demonstrated might have bad finite sample performance
in that consistent posterior for model selection necessarily implies the
maximal squared risk will be asymptotically larger than the optimal
$O(1/\sqrt{n})$ rate. This is the Bayesian version of the same phenomenon that
has been noted and studied by other authors.
