I use a Half Cauchy prior for the standard deviation parameter of a Normal distribution in a hierarchical model. I saw that in most PyMC3’s examples, the parameter of this distribution is fixed to \beta=0.5.
However, I guess that this choice is not arbitrary and that it must be related in some way to the value that we expect to find for parameter it models (in that case a standard deviation).
What is the best strategy to choose it? And does it impact the inference?