Gelman's (alpha + beta)^-5/2 parametrization for beta prior in AB test

That’s somewhat of an ugly parametrization; using

x = \frac{a}{a+b}
and
y = \sqrt{a+b}

you can sample x and 1/y uniformly, and use the transformation

a = xy^2
b = y^2 - xy^2 = (1-x)y^2

to get an equivalent prior from uniform samples. See Hierarchical AB testing in PyMC3 - #2 by chartl