Read this example from the book "Probability and statistics" by Morris H. DeGroot:
"Glove Use by Nurses. Friedland et al. (1992) studied 23 nurses in an inner-city hospital before and after an educational program on the importance of wearing gloves. They recorded whether or not the nurses wore gloves during procedures in which they might come in contact with bodily fluids. Before the educational program the nurses were observed during 51 procedures, and they wore gloves in only 13 of them. Let θ be the probability that a nurse will wear gloves two months after the educational program. We might be interested in how θ compares to 13/51, the observed proportion before the program.
We shall consider two different prior distributions for θ in order to see how sensitive the posterior distribution of θ is to the choice of prior distribution. The first prior distribution will be uniform on the interval [0, 1], which is also the beta distribution with parameters 1 and 1. The second prior distribution will be the beta distribution with parameters 13 and 38. This second prior distribution has much smaller variance than the first and has its mean at 13/51. Someone holding the second prior distribution believes fairly strongly that the educational program will have no noticeable effect.
Two months after the educational program, 56 procedures were observed with the nurses wearing gloves in 50 of them. The posterior distribution of θ, based on the first prior, would then be the beta distribution with parameters 1 + 50 = 51 and 1 + 6 = 7. In particular, the posterior mean of θ is 51/(51 + 7) = 0.88, and the posterior probability that θ > 2 × 13/51is essentially 1. Based on the second prior, the posterior distribution would be the beta distribution with parameters 13 + 50 = 63 and 38 + 6 = 44. The posterior mean would be 0.59, and the posterior probability that θ > 2 × 13/51is 0.95. So, even to someone who was initially skeptical, the educational program seems to have been quite effective. The probability is quite high that nurses are at least twice as likely to wear gloves after the program as they were before."
Now, I don't quite understand the logical reasoning behind that resulting comparison of the two posterior distributions (1st one being better than the 2nd). Let's say we consider a uniform prior distribution. It means that we think that it just happened so that there had been only 13 nurses who used gloves, and it could've just as easily happened that all of them, or none of them used gloves. In other words, the distribtuion is uniform. That means that after we get the posterior result, and it's very different, it carries relatively little information about whether the real distribtion changed a lot after the educational program, because even if the program didn't change anything at all, a result of 50/56 nurses wearing gloves would be just as expected as any other other, including the initial result. While if we consider a very pronounced prior distribution, then the fact that the result of experiment after the program is very far off would strongly suggest that the real distribution is now in fact very different. Where is the fault in my reasoning?