MCMC based results

We run the model on the basis of the published bare data summarized in Tab. . The MCMC results concerning the `efficacy parameter' $\epsilon$ are summarized in Tab. , to be compared with the published results, shown in Tab.  and repeated below our ones for the reader's convenience.

The MCMC based pdf's of $\epsilon$ are plotted in Fig.  with smooth curves showing the profile of the histograms of the $\epsilon$ values in the chains.

Figure: MCMC results for the vaccine efficacies. The vertical lines indicate the results provided by the pharma companies, in practical perfect agreement with the mode of the MCMC estimated probability distributions.

For comparison, the vertical dashed lines show the results of the pharma companies (`efficacy value' in Tab. ). As we can see, they correspond practically exactly to the modal values of the distributions. This makes us quite confident about the validity of our simple model for this quantitative analysis, although we maintain that the single number for the efficacy to be provided is not the mode, but rather the mean of the distribution, as we shall argue in Sec. . However, some remarks are in order already at this point. In fact, although there is no doubt about the fact that the most complete description of a probabilistic inference is given by the pdf of the quantity of interest, it is also well understood that it is often convenient to summarize the distribution with just a few numbers.

Usually, when inferring physical quantities, the preference goes to the mean and the standard deviation (the latter being related to the concept of standard uncertainty [14]) because of rather general probability theory theorems which make their use convenient for further evaluations (`propagations', as we shall also see in Sec. ). Other ways to summarize with just a couple of numbers a probability distribution are intervals which contain the uncertain value of the variable of interest at a given probability level (credible interval). We report then in Tab.  the 95% central credible interval <sup>7</sup>evaluated from the MCMC chains as well as the 90% `right side credible interval'. Other useful summaries, depending on the problem of interest, can be the most probable value of the distribution (mode) and the median, i.e. the value that divides the possible values into two equally probable intervals. As we have stated above, the modes of the MCMC based pdf's coincides with the values reported as `efficacy value' in Tab. , which contains also what we have generically indicated as 95% `uncertainty interval', in form of credible interval for Pfizer and confidence interval for the other two companies.⁸

The MCMC also provides results for the other `unobserved' nodes of the causal model, in our case $p_A$ and $n_{V_A}$. We refrain from quoting results on the `assault probability', because they could easily be misunderstood, as they strongly depend, contrary to $\epsilon$, on the values of $n_V$ and $n_P$, being $p_A$ a catch-all quantity embedding several real life variables, including the virus prevalence. We have however checked that our main results on $\epsilon$ are stable against the (simultaneous) variations of $n_V$ and $n_P$ by orders of magnitude (thus implying similar large variations of $p_A$).⁹

We give, instead, the results concerning $n_{V_A}$ that we expect to be around $n_{P_I}$. We get, in fact, respectively for Moderna-1, Moderna-2, Pfizer, AstraZeneca (LDSD) and AstraZeneca (SDSD) the following values: $89\pm 13$, $185\pm 19$, $160\pm 18$, $29\pm 8$ and $70\pm 12$ (note that the standard uncertainty is not simply the root square of $n_{V_I}$, as a rule of thumb would suggest).