The R binom library has confidence intervals function for the Bayes method that uses the Beta distribution. According to the binom documentation:
The default prior is Jeffreys prior which is a Beta(0.5, 0.5) distribution. Thus the posterior mean is (x + 0.5)/(n + 1).
p|x ~ Beta(x + prior.shape1, n - x + prior.shape2) The prior.shape1 and prior.shape2 can be passed in like so:
binom.bayes(x, n, conf.level = 0.95, type = c("highest", "central"), prior.shape1 = 0.5, prior.shape2 = 0.5, tol = .Machine$double.eps^0.5, maxit = 1000, ...)
Remembering that the default Bayes formula is (x + 0.5)/(n + 1) what would the shape parameters be to modify the 0.5 and 1 as i and j in the next formula.
(x + i)/(n + j)
such that i and j effectively makes the a different centralizing ratio rather than just merely 0.5.
The justification for this is that we may believe the actual mean is closer to say .3 rather than 0.5.
What are the rules to picking different shape parameters for differing prior knowledge i.e. 0.0 to 1.0 or 0% to 100%?
For example, for the Sun to rise tomorrow, we would have a priori knowledge of believe of 100% but for 10 billion years from now, it may be 80%.