# How does the predictive distribution affect Bayesian Online Changepoint Detection?

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.

• I'm not sure about this analytical approach, but you could take a computation approach using the R package mcp which 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. mcp supports AR(N) models, if that's sufficient. Commented Jan 28, 2021 at 12:51
• Unfortunately your package doesn't work for my use case, which is a) unknown number of change points and b) multivariate data. BOCD does work here. Commented Jan 29, 2021 at 2:06

It can not be 1-$$\pi$$ as $$\pi$$ is a pdf, not a pmf. I feel it should be $$\pi_0$$ for all terms in the summation. Because if $$r_t=0$$, the new observation is independent of any of the previously built models. It comes from a yet to be determined model. Thus the prior is the best guess we have irrespective of the previous run length