How do I specifya non-flat improper in brms

Motivation: I’m trying to verify that some disagreement between an rstan and brms analysis is the result the use of a flat prior for a variable in brms and a flat prior for a transformed version of that variable in my rstan code.

Issue: for a parameter x_0 I am estimating, I’d like to be able to specify a prior \pi where \pi(x0) \propto 1/(x_0 - x_{\max})^2. where x_0 is constrained as x_{\min} < x_0 < x_{\max} and x_{\min} > 0.

  1. Is there an easy way to do this in brms? I can’t find any documentation about such improper priors.

  2. Alternatively, is there a distribution that follows this formula?

I think this is a truncated Cauchy with scale parameter 1. Does that look right?

Thanks, that gives me a good place to start digging deeper. While I am aware of the Cauchy distn, I’ve never looked at it (or things other than the normal, log-normal, beta, dirichlet, gamma, inverse gamme, poisson, etc) carefully and so wasn’t able to see the connection.