Setting up the problem

Let us imagine we have a population of $N$ elements, a proportion $p$ of which shares a given character. The simplest example is that of a box containing $N$ balls, $n_1$ white and $n_2$ black. Let $p$ be the proportion of white balls, i.e. $p=n_1/N$. If we extract at random $m$ balls, then we roughly expect $m_1\approx n_1\times m/N = p\cdot m$ white and $m_2\approx n_2\times m/N = (N-n_1)\times m/N = (1-p)\cdot m$ black. A classical problem in probability theory is to infer the proportion $p$ from the observed (`measured') proportion $p_m = m_1/m$.

Obviously, if $m$ is equal to $N$, i.e. if we completely empty the box, then we acquire full knowledge of the box content and the solution is trivial. However, in most cases we are unable to analyze the entire population and we have to infer $p$ from a sample. Therefore, although $p_m$ can be a reasonable rough estimate of $p$, we can never be sure about the true proportion. At most, there are numerical values we shall believe more (those around $p_m$) and others we shall believe less. This problem was first tackled analytically by Laplace in 1774 [27].

Let us now complicate the problem, taking into account the fact that we are not even sure about the characteristics of each sampled individual, as, instead, it happens with black and white balls. This is exactly what happens with infections of different kinds, unless the symptoms are so evident and unique to rule out any other explanation. We have then to rely on tests that are typically not perfect, especially if we have neither time nor money to inspect in detail each individual in order to really see the active agent. Sticking to tests providing only a binary response,⁶as we hear and read in the media, and assuming that such testing devices and procedures are planned to detect the infected individuals, we expect that if the answer is positive then there should be a quite high chance that the individual is really infected, and a small chance that she is not. Similarly, if the answer is negative, there should be a high chance that the individual is not infected. (The conditionals are due to the fact that there are other pieces of information to take into account, as we shall see.)

We can characterize therefore the test by two virtually continuous numbers $\pi_1$ and $\pi_2$ in the range between 0 an 1 such that, depending on whether the individual is infected or not, the test procedure provides positive and negative answers with probabilities

$$P( \,\vert\, ) = \pi_1$$

$$P( \,\vert\, ) = 1-\pi_1\,;$$

$$P( \,\vert\, ) = \pi_2$$

$$P( \,\vert\, ) = 1-\pi_2\,,$$

with self-evident meaning of the symbols (we just remind that the `$\vert$' indicates that what follows it plays the role of conditions and therefore `$\vert$' should be read as “under the condition”, or “conditioned by”). More technically, $\pi_1$ is defined as test sensitivity, while $(1-\pi_2)$ is the test specificity (see e.g. Ref. [28]). Therefore, in order to fix the ideas, the test to which we are referring [16] has 98% sensitivity and 88% specificity.

As it is easy to understand, the numerical quantities of $\pi_1$ and $\pi_2$ do not come from first principles, but result from previous measurements. They are therefore affected by uncertainty as all results in measurements typically are [29]. Therefore, probability distributions have to be associated also to the possible numerical values of these two test parameters. Anyway, within this section we take the freedom to use their nominal values of 0.98 and 0.12 for our first rough considerations.