What is the probability that a vaccinated person gets shielded from Covid-19?

It is now time to come to the question asked in the title. We have already used the noun efficacy, associated to the uncertain variable $\epsilon$ of our model of Fig. . Then, analyzing the published data, we have got by MCMC several pdf's of $\epsilon$, that is $f(\epsilon\,\vert\,$Moderna-1$)$, $f(\epsilon\,\vert\,$Moderna-2$)$, and so on (see Fig. ). Hereafter, since what we are going to say is rather general, we shall indicate the generic pdf by $f(\epsilon\,\vert\,$data$,H)$, where $H$ stands for the set of hypotheses¹⁰underlying our inference and not specified in detail.

Let us now focus on the probability that an assaulted individual gets infected. Indicating by $A$ the condition `the individual is assaulted', by $V$ the condition `vaccinated' and by $I$ the event `the individual gets infected' (and therefore $\overline{A}$, $\overline{V}$ and $\overline{I}$ their logical negations), we get, rather trivially,

$$P(I\,\vert\,\overline{A}) = 0\,,$$ (4)

while, in the case of assault, the probability of infection depends on whether the individual has been vaccinated or not. In the case of placebo, following our model, we simply get

$$P(I\,\vert\,A,\overline{V}) = 1\,.$$ (5)

Instead, in case the individual has been vaccinated, the probability of infection will depend on $\epsilon$, that is, for the special cases of perfect shielding and no shielding (i.e. no better than the placebo),

$$P(I\,\vert\,A,V,\epsilon=1) =$$ 0 (6)

$$P(I\,\vert\,A,V,\epsilon=0) = 1\,.$$ (7)

In general, if we were certain about the precise value of $\epsilon$, the probability of getting infected or not is related in a simple way to $\epsilon$:

$$P(I\,\vert\,A,V,\epsilon) = 1-\epsilon$$ (8)

$$P(\overline{I}\,\vert\,A,V,\epsilon) = \epsilon\,.$$ (9)

The above equations, and in particular Eq. (), express in mathematical terms the meaning we associate to efficacy, in terms of the model parameter $\epsilon$: the probability that a vaccinated person gets shielded from a virus (or from any other agent). But the value of $\epsilon$ cannot be known precisely. It is, instead, affected by an uncertainty, as it (practically) always happens for results of measurements [14] (and indeed also the pharma companies accompany their results with uncertainties – see Tab.). In a probabilistic approach, this means that there are values of $\epsilon$ we believe more and values we believe less. All this, we repeat it, is summarized by the probability density function

$$f(\epsilon\,\vert\,$$data$$,H)\,.$$

The way to take into account all possible values of $\epsilon$, each weighted by $f(\epsilon\,\vert\,$data$,H)$, is to follow the rules of probability theory, i.e.

$$P(\overline{I}\,\vert\,A,V, ,H) = \int_0^1\!P(\overline{I}\,\vert\,A,V,\epsilon)\cdot f(\epsilon\,\vert\, ,H)\, \epsilon\,.$$

Using then Eq. () we get

$$P(\overline{I}\,\vert\,A,V, ,H) = \int_0^1\!\epsilon\cdot f(\epsilon\,\vert\, ,H)\, \epsilon\,,$$ (10)

which represents the probability that a vaccinated person, not belonging to the trial sample, gets shielded from Covid-19, on the basis of the data obtained from the trial and all (possibly reasonable) hypotheses assumed in the data analysis. It is easy to understand that $P(\overline{I}\,\vert\,A,V,$data$,H)$ is what really matters and therefore what should be communicated as efficacy to the scientific community and to the general public.¹¹

Now, technically, Eq. () is nothing but the mean of the distribution of $\epsilon$. This should then be the number to report, and not the mode of the distribution, which has no immediate probabilistic meaning for the questions of interest.¹²

Now, if we compare the `efficacy values' of Tab.  with the mean values of Tab.  we see that in most cases the differences are rather small (about 1/3 to 1/2 of a standard deviation), although the modal values (to which, as we have showed above, the published efficacies correspond) are always a bit higher than the mean values, due to the left skewness of the pdf's. Therefore our point is mostly methodological,¹³ with some worries when the mean value and the most probable one differ significantly.