I have a continuously growing list of probability estimations of an event $$[p(e,t1),p(!e,t2),\ldots,p(e,tn)]$$, where $p(e,t)$ is the probability of the event $e$ or not $e$ happening at time t. I't like to have a random variable that models these estimations.
A beta distributed variable does not seem to be the right choice, because as far as I understand it, it is supposed to model binary observations, not real values in the $[0,1]$ interval.
So if I observe a $p(e,t2)=0.4$, and my beta distributed variable is $B(\alpha=1, \beta=1)$, one option would be to update $\alpha\leftarrow\alpha+0.4$, and $\beta\leftarrow\beta+0.6$. I have the feeling I am being too creative though, because the choice of scale seems to be arbitrary (i.e. why not $\alpha\leftarrow\alpha+4$, and $\beta\leftarrow\beta+6$ for example) ?
Would this approach be correct? If not, what should be the correct choice here?