I am looking for a prior for a scale parameter for which I have prior knowledge such that: "$\sigma$ typically does not exceed 100." ("typically" meaning that occasionnally this can happen).
In such a context, I notice in the paper "Prior distributions for variance parameters in hierarchical models" of Andrew Gelman the following recommandation:
[...] When more prior information is desired, for instance to restrict σ away from very large values, we recommend working within the half-t family of prior distributions, which are more flexible and have better behavior near 0, compared to the inverse-gamma family. A reasonable starting point is the half-Cauchy family, with scale set to a value that is high but not off the scale."
As I understand it, a Cauchy (thus half-Cauchy) distribution has an infinite variance and I am not confortable with the idea of building an informative prior with an infinite variance density. Have you some insight on why my interpretation is bad/unsuited ? Moreover, have you some alternative proposals for my prior ?