The posterior distribution of run length P (r t |x 1:t ) can be computed recursively: where: The joint distribution over run length r t and data x 1:t can be derived by summing P (r t , r t−1 , x 1:t ) over r t−1 : This formulation updates the posterior distribution of the run length given the prior over r t from r t−1 and the predictive distribution of new data.
the existing BO-CPD model (Adams and MacKay, 2007) specifies the conditional prior on the change point P (r t |r t−1 ) in advance.