AB Testing Hierarchical Model -- Case Study Question


In the case study Partial Pooling (model #3) a prior is given as

parameters {
  real<lower=0, upper=1> phi;         // population chance of success
  real<lower=1> kappa;                // population concentration
  vector<lower=0, upper=1>[N] theta;  // chance of success 
model {
  kappa ~ pareto(1, 1.5);                        // hyperprior
  theta ~ beta(phi * kappa, (1 - phi) * kappa);  // prior
  y ~ binomial(K, theta);                        // likelihood

Is this prior a superior implementation of the one discussed in Gelman chapter 5 and shown in various blogs as suggested for AB testing of proportions to alleviate multi-comparison issues (e.g. click through rates for websites? psi is uniform in Stan by not assigning a prior?

If so…is there a reason why in this case study, the prior on theta is not defined using the full conversion in the beta from psi and kappa to alpha and beta? I was under the impression (and testing seems to support) that if it is not set up as below, then psi will not represent the overall average proportion.

Instead of …

beta(phi * kappa, (1 - phi) * kappa)


beta(phi * (kappa-2)+1, (1 - phi) * (kappa-2) +1))