Beta approximation of the MCMC results and its utility

Moving to our results about the `model parameter $\epsilon$' (it is now time to be more careful with names), reported in Tab.  and Fig. , it should now be clear why the number to report as efficacy should be the mean of the distribution. As far as the distribution of $\epsilon$ is concerned, given the similarity of the inferential problem that was first solved by Bayes and Laplace, we have good reasons to expect that it should not `differ much' from a Beta. In order to test the correctness of our guess we have done the simple exercise of superimposing over the MCMC distributions of Fig.  the Beta pdf's evaluated from mean and standard deviation of Tab. . The distribution parameters can be in fact obtained solving Eqs. () - () for $r$ and $s$:¹⁶

$$r = \frac{(1-\mu)\cdot \mu^2}{\sigma^2} - \mu$$ (23)

$$s = \frac{1-\mu}{\mu}\cdot \left[\frac{(1-\mu)\cdot \mu^2}{\sigma^2} - \mu\right]\,.$$ (24)

Figure: MCMC inferred distributions of $\epsilon$ (solid lines exactly as in Fig. ) with superimposed (dashed lines, often coinciding with the solid ones) the corresponding Beta distributions evaluated from the mean values and the standard deviations resulting from MCMC.

The result is shown in Fig. . As we can see, the agreement is rather good for all cases, especially for Moderna and Pfizer, for which the Beta and MCMC curves practically coincide.