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)
```

This:

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