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 hypotheses⁹underlying 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 [12] (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, thus obtaining

$$P(\overline{I}\,\vert\,A,V, ,H) = \int_0^1\!\epsilon\cdot f(\epsilon\,\vert\, ,H)\, \epsilon\,.$$ (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 that should therefore be communicated as efficacy to the scientific community and to the general public.¹⁰

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.

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, with some worries when the mean value and the most probable one differ significantly.