  We consider Bayesian analysis of a class of multiple changepoint models.
While there are a variety of efficient ways to analyse these models if the
parameters associated with each segment are independent, there are few general
approaches for models where the parameters are dependent. Under the assumption
that the dependence is Markov, we propose an efficient online algorithm for
sampling from an approximation to the posterior distribution of the number and
position of the changepoints. In a simulation study, we show that the
approximation introduced is negligible. We illustrate the power of our approach
through fitting piecewise polynomial models to data, under a model which allows
for either continuity or discontinuity of the underlying curve at each
changepoint. This method is competitive with, or out-performs, other methods
for inferring curves from noisy data; and uniquely it allows for inference of
the locations of discontinuities in the underlying curve.
