Notice that the posterior distribution looks a little different depending on what prior you use. The most common criticism leveled against Bayesian methods is that the choice of prior adds an unsavory subjective element to analysis. To a certain extent, they're right about the added subjective element, but their allegation that it is unsavory is way off the mark.
To see why, check out the following figure, which shows both posterior distributions (from priors #1 and #2) in the same plot. Notice how priors #1 and #2-two very different priors-given the evidence, produce posteriors that look more similar to each other than the priors did:
Figure 7.6: The posterior distributions from prior #1 and #2
Now direct your attention to the following figure, which shows the posterior of both priors if the evidence included 80 out of 120 correct trials:
Figure 7.7: The posterior distributions from prior #1 and #2 with more evidence
Note that the evidence still contains 67% correct trials,...