# What's the posterior distribution for vaccine effectiveness, given Pfizer's 12-15yo study results?

Summary: the placebo group had 1129 subjects and 18 infections. The vaccine group had 1131 subjects and 0 infections. More info here: https://www.pfizer.com/news/press-release/press-release-detail/pfizer-biontech-announce-positive-topline-results-pivotal

I'm interested in examining Pfizer's claim (same link) that this is "100% efficacy". Of course, one interpretation of that phrase is "nobody in the vaccinated group got COVID," which is trivially true, because that was directly observed. But I think people would tend to interpret it as something more like, "based on the study results, if I get the vaccine, my probability of getting COVID will be reduced by 100% (i.e. to 0%)." No finite sequence of vaccinated people not getting COVID can prove that vaccinated people get COVID exactly 0% of the time, so that doesn't sound right to me.

What is the posterior distribution for efficacy, given Pfizer's data, and given the definition "expectation of COVID probability reduction due to the vaccine" for "efficacy"? I'm taking the prior distribution for efficacy as the uniform distribution on [0,1]. Is that a reasonable prior distribution to use, in this context?

In case it's helpful in calibrating answers: I have a BS in math that covered more calc and algebra, and not so much statistics, and I'm currently working through Probability Theory by Jaynes in my free time. My intuition here is that the posterior distribution is the beta distribution with shape parameters 1 and 19.032, but I'm not confident in it.

That leaves a posterior of $$1140\theta^{1139}$$. All values with any density are quite close to 100%. The 99.99% interval is roughly [.99596,1].