Uncertainty about $P($Inf$\,\vert\,$Pos$)$ and $P($NoInf$\,\vert\,$Neg$)$?

As we have seen, the probabilities of interest, taking into account all the possibilities of $\pi_1$, $\pi_2$ and $p$ are obtained as weighted averages, with weights equal to $f(\pi_1,\pi_2,p)$. One could then be tempted to evaluate the standard deviation too, attributing to it the meaning of `standard uncertainty' about $P($Inf$\,\vert\,$Pos$)$ and $P($NoInf$\,\vert\,$Neg$)$. But some care is needed. In fact, although is quite obvious that, sticking again to $P($Inf$\,\vert\,$Pos$)$, we can form an idea about the variability of $P($Inf$\,\vert\,$Pos$,\pi_1,\pi_2,p)$ varying $\pi_1$, $\pi_2$ and $p$ according to $f(\pi_1,\pi_2,p)$ (something like we have done in Tab. , although we have not associated probabilities to the different entries of the table), one has to be careful in making a further step. The fact that the weighted average is $P($Inf$\,\vert\,$Pos$)$ comes from the rules of probability theory, namely from Eq. (), but there is not an equivalent rule to evaluate the uncertainty of $P($Inf$\,\vert\,$Pos$)$.

In order to simplify the notation, let us indicate in the following lines $P($Inf$\,\vert\,$Pos$)$ by $\cal P$. In order to speak about standard uncertainty of $\cal P$, we first need to define the pdf $f(\cal P)$, and then evaluate average and standard deviation. But Eq. () does not provide that, but only a single number, that is $\cal P$ itself.

Let us reword what we just stated using a simple example. Given the `random variable' $X$ and the pdf associated to it $f(x)$, mean and standard deviation of $f(x)$ provide expected value (`$\mu_X$') and standard deviation of $X$, and not of $\mu$.