In the case of the Beta, the obvious choice for reparameterization is in terms of a
mean parameter
\phi = \alpha /(\alpha + \beta)
and total count parameter
\lambda = \alpha + \beta
in the Stan user guide, the count parameter receives a Pareto prior with p(λ) ∝ λ ^{−2.5}.
however, Claassen (2019) used a Gamma prior of Gamma(4,0.1) and other Gamma prior of Gamma(3,0.04) in its case-study version at http://mc-stan.org. The post titled " Prior Choice Recommendations" don’t discuss this topic.
So, how should one choose a prior for the count parameter?
Claassen C (2019) Estimating smooth country-year panels of public opinion. Political Analysis 27 (1): 1-20. 10.1017/pan.2018.32