Which priors?

After having read in the first part of the paper the dramatic role of the prior, when we had to evaluate the probability of individual being infected, given the test result, one might be surprised by the regular use of a flat prior of $p$ throughout the present section. First at all, we would like to point out that we are doing so, in this case, not “in order to leave the data to `speak' by themselves”, as someone says. It is, instead, the other way around: the values of $p$ preferred by the data, starting from a uniform prior, are characterized by a distribution much narrower than what we could reasonably judge, based on previous rational knowledge. In other words, they are not at odds with what we could believe independently of the data. But this is not always the case, and experts could have more precise expectation, grounded on their knowledge.

Anyway, a prior distribution is something that we have to plug in the model, if we want to perform a probabilistic inference. In practice - and let us remind again that “probability is good sense reduced to a calculus” - we model the prior in a reasonable and mathematically convenient way, and the Beta distribution is well suited for this case, also due to the flexibility of the shapes that it can assume, as seen in Sec. . Once we have opted for a Beta, a uniform prior is recovered for $r=1$ and $s=1$, although we are far from thinking that $p=0$ or $p=1$ are possible, as well as that $p$ could be above 0.9 with 10% chance, and so on.