Solved – priors for Gamma shape and scale parameters

bayesianpriorstan

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?

Best Answer

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.

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