I had a question regarding how to best perform "sensitivity analysis" on a Beta prior used for a estimating a proportion from a binomially distributed data problem (i.e., beta-binomial problem).
I understand that the shape parameters $\alpha$ and $\beta$ move the Beta to right and left. But how much and on what basis should I change these two parameters and compare their effects on the the posterior 95% credible Intervals (picture below)?
(Note: In the picture below, 'A' parameter denotes 'alpha')