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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%.

Update

Here is a graph of the various Beta histograms: Beta distributions

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    $\begingroup$ You should make sure you understand the difference between a confidence and a credible interval. The difference will also impact how you choose the parameters. $\endgroup$ – jaradniemi Aug 3 '16 at 16:57
  • $\begingroup$ Person concerned was undoubtedly Jeffreys. If binom documentation spells otherwise, it's spelling wrongly. $\endgroup$ – Nick Cox Aug 4 '16 at 22:14
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The rules for choosing the shape parameters (hyperparameters) are those rules that apply to any prior choice: choose them in such a way that the prior distribution reflects your prior knowledge.

If your prior knowledge says that most of the prior probability should be concentrated around 75%-85% with mode at 80%, then you could choose the hyperparameters $Beta(800,200)$, which has mode at $\approx 0.8$ and cumulates $99.9\%$ of the mass in the interval $(0.75,0.85)$.

The choice of the Jeffreys priors is completely different since it is based on a formal criterion to produce a noninformative prior.

Comment: The use of "confidence interval" in the R package binom is rather unfortunate in this case since they actually calculate posterior credible intervals based on the beta prior which happen to have good frequentist properties. See: https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval

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  • $\begingroup$ Oh I see, the Beta parameter dictate a mean but the kurtosis is vastly different. (I added an image above.) Do you have a recommend read on this topic? $\endgroup$ – Chris Aug 4 '16 at 21:22
  • $\begingroup$ obscureanalytics.com/2012/07/04/… is a good read $\endgroup$ – Chris Aug 5 '16 at 3:11

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