Taking into account `informative priors' about $\epsilon$

The fact that the MCMC results are described with high degree of accuracy by a Beta distribution is not only a sterile curiosity, but has indeed an interesting practical consequence.

As we have seen, the pdf's of $\epsilon$ have been obtained starting from a uniform prior. The same must be for Pfizer, since they also did a Bayesian analysis, as explicitly stated in their paper [4] and as revealed by the expression `credible interval' (see Tab. ), and their values practically coincide with ours. Instead, in the case of the other results, the expression `confidence interval' seems to refer to a frequentistic analysis, in which “there are no priors”. But in reality it is not difficult to show that sound frequentist analyses (e.g. those based on likelihood) can be seen as approximations of Bayesian analyses in which a flat prior was used (see e.g. Ref. [14]). The resulting `estimate' corresponds to the mode of the posterior distribution under that assumption.

The question is now what to do if an expert has a `non flat' informative prior (indeed none would à priori believe that values of $\epsilon$ close to zero or to unity would be equally likely!). Should she ask to repeat the analysis inserting her prior distribution of $\epsilon$? Fortunately this is not the case. Indeed, as we have discussed in Ref. [17], due to the symmetric and peer roles of likelihood and prior in the so called Bayes' rule, each of the two has the role of `reshaping' the other. Moreover, since a posterior distribution based on a uniform prior concerning the variable of interest can be interpreted as a likelihood (besides factors irrelevant for the inference), we can apply to it an expert's prior in a second time (see Ref. [17] for details). It becomes then clear the importance of the observation that the pdf's of $\epsilon$ derived by MCMC can be approximated by Beta distributions: from the MCMC mean and standard deviation we can evaluate the Beta of interest, as we have seen above; this function can be then easily multiplied by the expert's prior; the normalization can be done by numerical integration and finally the posterior distribution of $\epsilon$ also conditioned on the expert's prior can be obtained.

This implementation in a second step of the expert opinion becomes particularly simple if also her prior is modeled by a Beta, recognized to be a quite flexible distribution. For example, indicating by $r_F$ and $s_F$ the Beta parameters $[$calculated with Eq. () - ()$]$ obtained by a flat prior and by $r_0$ and $s_0$ the parameters of the Beta informative prior, the posterior distribution will still be a Beta with parameters¹⁴

$$r_p = r_0 + r_F -1$$ (25)

$$s_p = s_0 + s_F -1\,.$$ (26)