I was wondering if anyone here knew of any recommendations for the “certainty” or “kappa” parameter of a Beta distribution?

From these two sources:

- http://mc-stan.org/users/documentation/case-studies/pool-binary-trials.html
- https://docs.pymc.io/notebooks/hierarchical_partial_pooling.html

the default non-informative prior is a Pareto(alpha=1, m=1.5) distribution, but using this distribution results in tails that are too long for my purposes.

Is there any other distribution that would be recommended as the prior for this parameter that has shorter tails? Basically, the resulting inference has certainty parameters on the order of thousands when the data themselves have sample sizes on the order of tens and I want to incorporate this information into a prior.