I'm currently reading the Probabilistic Programming and Bayesian Methods for Hackers "book". I've read a few chapters and I was thinking on the first Chapter where the first example with pymc consist of detecting a witchpoint in text messages. In that example the random variable to indicate when the switchpoint is happening is indicated with $\tau$. After the MCMC step the posterior distribution of $\tau$ is given:
Firstly what can be learned from this graph is that there is a propability of almost 50% that the switchpoint happend on day 45. Though what if there wasn't a switchpoint ? Instead of assuming there is a switchpoint and then trying to find it, I want to detect if there is in fact a switchpoint.
The author answers the question "did a switchpoint happen" by "Had no change occurred, or had the change been gradual over time, the posterior distribution of $\tau$ would have been more spread out". But how can you answer this with a propability, for example there is a 90% chance a switchpoint happend, and there is a 50% chance it happend on day 45.
Does the model need to be changed ? Or can this be answered with the current model ?