The log-likelihood function is
- 1.choose one ofat random, obtaining
- 2.propose a change tosuch that.
- 3.the acceptance probability isNote thatit follows that the CDF ofisandsoThen we have
- 1.Draw one ofat random, obtaining say.
- 3.The acceptance probability issince
Choose a position
uniformly distributed on
, which must lie within an existing interval
Propose new heights
for the step function on the subintervals
. Use a weighted geometric mean for this compromise,
and define the perturbation to be such that
drawn uniformly from
The prior ratio, becomes
the proposal ration becomes
and the Jacobian is
is removed, the new height over the interval
, 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