Expected value or most probable value of $\pi_1$ and $\pi_2$?

At this point someone would object that one should use the most probable values of $\pi_1$ and $\pi_2$, rather than their expected values. The answer is rather simple. Let us consider again Eq. (). Assuming a well precise value of $\pi_1$, the probability of Positive if Infected is exactly equal to $\pi_1$. However, if we want to evaluate $P($Pos$\,\vert\,$Inf$)$, taking into account all possible values of $\pi_1$ and how much we believe each of them, that is $f(\pi_1)$, we just to need to use a well known result of probability theory:

$$P( \,\vert\, ) = \int_0^1\!P( \,\vert\, ,\pi_1)\cdot f(\pi_1)\, \pi_1\,.$$ (37)

But, being $P($Pos$\,\vert\,$Inf$,\pi_1) = \pi_1$, we get

$$P( \,\vert\, ) = \int_0^1\!\pi_1\cdot f(\pi_1)\, \pi_1\,,$$ (38)

in which we recognize the expected value of $\pi_1$.²¹