This is a belated continuation of the post on the Beta-Binomial distribution.
Firstly, if we set a=b=1, then
In this instance, is equally likely prior to observing ANY data where the prior information is a flat Beta (1,1) distribution. The interest in this comes from the fact that this is what Bayes' verbal exposition of his argument was based on. In the early 1700's, Laplace proved this result mathematically.
Other non-informative priors also exist.
For instance, if we take a=b=0.5, then:
This instance is NOT constant. As n=1,2,3,... increases we see the distribution become a U shaped plot.
The distribution moves more and more towards a Beta(0.5, 0.5) density, but between 0 and n. This implies an expectation of more data close to 0 or n and little (if any) in the centre of the density plot (that is between 0 and n but not 0 or n). We expect this as the Beta(0.5, 0.5) density produces a higher probability close to and .