In Bayesian Online Changepoint Validation (1), we try to segment a time series by changepoints. The crux of the algorithm, I think, is calculating the probability that a new datum has experienced a changepoint versus being part of the previous run, which intuitively should rely on the predictive probability $\pi_t = P(\mathbf{x_t} \mid \theta)$. This is calculated in the format algorithm, which I have included from the paper here:
However what confuses me is that both the changepoint probability $P(r_t=0, \mathbf{x_{1:t}})$ and the growth probability $P(r_t=r_{t-1}+1, \mathbf{x_{1:t}})$ are both multiplied by this predictive probability $\pi_t$. Considering this, how is the predictive probability affecting anything in the algorithm? Couldn't it be removed as a common factor? I clearly don't understand the algorithm very well, but I would have expected the changepoint probability to be proportional to $1-\pi_t$ since the less likely the incoming data point, the more likely we have encountered a changepoint.
(1) Adams, R.P. and MacKay, D.J., Bayesian online changepoint detection. arXiv 2007. arXiv preprint arXiv:0710.3742.
mcpwhich returns posteriors for the change points (and other parameters). The probability that $x_i$ is part of segment $k$ is then just the cumulative density of the change point posterior for $changepoint_{k-1}$. I can write up a worked example if this could be an acceptable answer.mcpsupports AR(N) models, if that's sufficient. $\endgroup$