More on the probability that a vaccinated person gets shielded from the virus – a Monte Carlo approach

Section has been devoted to explain the reason why the proper number to be reported as `efficacy', meant as the probability that a vaccinated person is shielded from the virus, is the mean of the pdf of the model parameter $\epsilon$. Having talked in this section about predicting the number of infects, we can use the same extended model of Fig.  in order to check the outcome of that reasoning. It is in fact enough to set $n'_V=1$ and $p'_A=1$ and analyze the result of the MCMC. Indeed, $n'_{V_A}$ will be identically 1, in the sense that the person will be `assaulted' with certainty, but the output $n'_{V_I}$ can have now only two possible values, 0 (person not infected) or 1 (person infected). If $\epsilon$ were exactly known, and let us indicate it by $\epsilon_K$, the probability of $n'_{V_I}=0$ would be $\epsilon_K$ and that of $n'_{V_I}=1$ would be $1-\epsilon_K$. A simple direct Monte Carlo would then produce a fraction of occurrences of $n'_{V_I}=0$ around $\epsilon$. If, instead, $\epsilon$ is unknown, then the values used in the bottom-left side of the network will be those occurring in the MCMC chain. Therefore, we reobtain the same result seen in Sec., i.e. that the relative frequency of the occurrence of $n'_{V_I}=0$ will be equal to the mean of $\epsilon$ in the chain.

This MCMC strategy offers a further argument, to which some practitioners might be more reactive, in support of the thesis that the number to be reported as `efficacy' should be the average of the probability distribution of $\epsilon$, rather than other possible summaries of the distribution.