Results

We run the model assuming that the Moderna claims come from the entire sample, i.e. about thirty thousand people, equally divided between placebo and vaccine groups. This assumption is, anyway, not relevant for the efficacy estimates, as we have verified running the same model for samples one-tenth and one-hundredth of the full size and getting practically the same results for the efficacy.

The results of the MCMC sampling are reported in Fig. 2

Figure: MCMC results for the vaccine efficacies (see text).

with smooth curves that follow the profile of the histograms of the MCMC `data' (one million steps have been chosen in order to reduce the sampling fluctuations): the black one (a bit broader) for Moderna; the blue one (a bit narrower) for Pfizer. The vertical dashed lines show the press release results of the two companies, that is 0.945 and 0.95, respectively. Indeed they correspond `practically exactly' to the modal values of the distributions. But this is only one possible summary of a distribution, and not always the best one, especially if not associated to an uncertainty (and, certainly, the `$p <0.0001$' given by both companies does not provide such information).

Usually our preference goes to the mean and the standard deviation because of rather general Probability Theory theorems, which make their use convenient for further evaluations (to this standard deviation is related the concept of standard uncertainty [3]). Other ways to summarize with just a couple of number a probability distribution is the `central' interval which contain the uncertain variable of interest at a given probability level credible interval. We report in the following table these summaries, that provide a quantitative evaluation of the uncertainty, together to the probability that the `true value' of the efficacy is larger than 90%, reminding however that the most complete, quantitative information of the inference is contained in the curves of Fig. 2:

	mean $\pm$ stand. unc.	centr. 95% cred. int.	$P(\epsilon \ge 0.9)$
Moderna	$0.933 \pm 0.029$	$[0.866, 0.976]$	0.872
Pfizer	$0.944 \pm 0.019$	$[0.900, 0.975]$	0.976

We would like to remind that this results do not depend on the exact values of $n_V$ and $n_P$, provided they are enough larger than $n_{P_I}$.

As it is easy to expect, the MCMC also provides results on the other `unobserved' nodes of the causal model, in our case $p_A$ and $n_{V_A}$. We refrain to quote results on the `assault probability', because they could easily be misunderstood, as they strongly depend, contrary to $\epsilon$, on the precise values of $n_V$ and $n_P$, being $p_A$ a catch-all quantity embedding several real life variables, including the virus prevalence. We give, instead, the results concerning $n_{V_A}$, weakly dependent on $p_A$ and $n_{V_A}$ and that we expect to be of the order of magnitude of $n_{V_I}$. We get, in fact, respectively for Moderna and Pfizer, $89\pm 13$ and $161\pm 18$ (note that the standard uncertainties are not simply $\sqrt{n_{V_I}}$, as a rule of thumb would suggest).