Fixed Dimension Reassessment
The log-likelihood function is
choose one of at random, obtaining
propose a change to such that .
the acceptance probability is
it follows that the CDF of is
Then we have
Draw one of at random, obtaining say .
The acceptance probability is
Choose a position uniformly distributed on , which must lie within an existing interval w.p 1.
Propose new heights for the step function on the subintervals and . Use a weighted geometric mean for this compromise,
and define the perturbation to be such that
with drawn uniformly from .
The prior ratio, becomes
the proposal ration becomes
and the Jacobian is
If is removed, the new height over the interval is , the weighted geometric mean satisfying
The acceptance probability for the corresponding death step has the same form with the appropriate change of labelling of the variables, and the ratio terms inverted.
The histogram of number of change points is
And we can get the density plot of position (or height) conditional on the number of change points , for example, the following plot is for the position when