Introduction

Our perspectives about living with Covid-19 are changed dramatically in just a few days with the results from vaccine trials of the past days by Pfizer and Moderna. The former claimed a 90% efficacy (then updated to 95%); the latter 94.5%. Obviously, the media did not mention any uncertainty, so we understood that the initial Pfizer's number was the result of a rounding, with uncertainty of the order of the percent. We were more surprised by the Moderna's one, providing the tenths of the percent, as if it were much more precise. Looking around, we had the impression that the “point five” was taken very serious, not only by media speakers, who put the emphasis on the third digit, but also by experts from which we would have expected a phrasing implying some uncertainty in the result (see e.g. Ref.[1]).

A fast exercise showed that, in order to have an uncertainty of the order of a few tenths of percent, the number of vaccine-treated individuals that got the Covid-19 had to be at least of the order of several hundreds. But this was not the case. In fact, the actual numbers were indeed much smaller: “This first interim analysis was based on 95 cases, of which 90 cases of COVID-19 were observed in the placebo group versus 5 cases observed in the mRNA-1273 group, resulting in a point estimate of vaccine efficacy of 94.5% ($p <0.0001$)”[2]. Now, it is a matter of fact that if a physicist reads a number like `5', she tends to associate to it, as a rule of thumb, an uncertainty of the order of its square root, that is $\approx 2.2$. Applied to the Moderna claims, this implies an inefficacy of $\approx (5.5\pm 2.3)\%$, or an efficacy of $\approx (94.5\pm 2.3)$. Another reason to worry about the scientific validity of the result [3] was not only the absence of an uncertainty associated to the result, but also the “$p <0.0001$” accompanying it, especially for those who are extremely critical against p-values and other frequentist prescriptions [4].

We then tried to see whether it was possible to get an idea of the possible values of efficacy consistent with the data, each associated with a degree of belief. In other words, we have tried to critically review the claims, on the basis of the scarce data made available, following a sound probabilistic approach, in order to arrive to a probability density function (pdf), although not obtained in closed form, of the quantity of interest. In these kinds of situations, we have learned (see e.g. [5]) that the most important starting point is to build up a graphical representation of the causal model relating the quantities of interest, some of them `observed' and others `unobserved', among the latter the quantities we want to infer. Also in this case, despite some initial skepticism about the possibility to get some reasonable results, because of the scarce information in our hands, once we have built up the model, very basic indeed, it was clear that the main result about the efficacy was not depending on the many details of the trials.