priors for Gamma shape and scale parameters

I have a random variable $X$ that is Gamma distributed with unknown parameters $\alpha$ and $\beta$: $$X\sim \text{Gamma}(\alpha, \beta)$$

I now want to estimate $\alpha$ and $\beta$ from samples $x_i$ in a Bayesian setting using STAN. What is the recommended (weakly informative) prior distribution for these parameters? In STAN $\alpha$=shape and $\beta$=inverse scale.

I read that the half-Cauchy distribution should be used for scale parameters, so is

shape ~ cauchy(0,2.5);
scale ~ cauchy(0,2.5);

the recommended prior for both?

Alternatively, the reference prior for the ordering $\alpha$, $\beta$ is (http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf page 13): $$\Pi(\alpha,\beta) \propto \frac{\sqrt{\alpha PG(1,\alpha)-1}}{\sqrt{\alpha}\beta}$$ where $PG(1,x)=\sum_{i=0}^{\infty} (x+ i)^{-2}$ is the polygamma function. It results in proper posteriors.
• Watch out, they use the location / scale parametrization but denote it with $\alpha$ and $\beta$. – jgyou Feb 12 at 21:42
Half-Cauchy family was recommended instead of the inverse-gamma prior for scale parameters. Gelman (2006) recommended this because the inverse-gamma prior could be sensitive in inference problems if the variance estimates are close to zero. The density function of half-Cauchy is as follows (it only takes one parameter, $d$): $$f(x|d) = \frac{2d}{\pi\left(d^{2} + x^{2}\right)}, \quad x>0, d>0$$ Therefore, you can use half-Cauchy(2.5) for parameters greater than zero.