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:

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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.

  • $\begingroup$ 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. $\endgroup$ Commented Jan 28, 2021 at 12:51
  • $\begingroup$ 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. $\endgroup$
    – Migwell
    Commented Jan 29, 2021 at 2:06

1 Answer 1


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


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