Conclusion

A re-analysis of the 2020 data on vaccine trials by Pfizer, Moderna and AstraZeneca has been performed with didactic intent and focusing on uncertainties and on what most effectively should be reported as outcome. With this respect, we appreciate once more the role provided by Bayesian networks in clarifying the meaning of each variable entering the model and its relation with the others. In particular, we make a distinction between the `efficacy variable' $\epsilon$ and the efficacy to be reported to the scientific community and to the general public as probability that a newly vaccinated person will shielded from Covid-19. The uncertain values of $\epsilon$ are characterized by the pdf $f(\epsilon)$, obtained in this work by MCMC, and whose mean value has exactly the meaning of efficacy, in analogy to Laplace's rule of succession to quantify the probability of a future event.

We have stressed not only the importance of providing the most complete information of $f(\epsilon)$, whose graphical representation provides better than many words its uncertainty, but also that of summarizing the results (if only two numbers have to be chosen) with mean and standard deviation of the distribution. In fact, although the probability distribution of $\epsilon$ is what is really needed to the development of predictive what-if scenarios, mean and standard deviation are the most useful quantities to be used for further approximated evaluations.

With regards to the comparison with the published result, the efficacy values obtained in this analysis as mean of the probability distribution of $\epsilon$ are in good agreement with them for the reasons just reminded in the previous section, with the exception of the 100% claim that speaks for itself. This agreement plays an important role in validating the causal model on which the present analysis is based, indeed very simple (but not simplistic!), that can be used by students and researchers to repeat the analysis – for this reason pieces of programming code have also been provided.

Note added
We have learned in the meanwhile that the strategy of choosing the vaccine sample much larger than the placebo one, which we suggest at the end of the previous section, has been used in real life trials by Sputnik [2] and AstraZeneca [3], which used vaccine vs placebo sample ratios of 3:1 and 2:1, respectively.