# Gamma univariate

Suppose ${\mathbf {x}}=(x_{1},x_{2},\cdots ,x_{n})$ where xi are iid observations from the gamma distribution with density

$Ga(x|\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x},\,x>0;\,(\alpha >0,\beta >0).$ In the parameterization above $\alpha$ represents shape while $\beta$ is a rate parameter. An alternative parameterization is given by $(\alpha ,\theta =1/\beta )$ , and in that case $\theta$ becomes a scale parameter. Another possible parameterization is $(\alpha ,\mu )$ , where${\textstyle \mu =E(x|\alpha ,\beta )=\alpha /\beta ,}$ so that $\alpha$ and $\mu$ are orthogonal (the Fisher information is diagonal).

In what follows, $\psi _{1}(\alpha )={\frac {d^{2}}{d\alpha ^{2}}}\log(\Gamma (\alpha ))$ is the trigamma function.

Overall Recommended (OR) prior
$\pi (\alpha ,\beta )={\frac {\sqrt {\alpha \psi _{1}(\alpha )-1}}{{\sqrt {\alpha }}\beta }}$ In other parameterization
$\pi (\alpha ,\mu )=\mu ^{-1}{\sqrt {\frac {\alpha \psi _{1}(\alpha )-1}{\alpha }}}$ Recommended prior if $\beta ~$ is of main interest
$\pi (\alpha ,\beta )={\frac {\sqrt {\psi _{1}(\alpha )}}{\beta }}$ ### Formal endorsements

The OR prior is the reference prior for several ordering of the parameters including $\{\alpha ,\beta \},\{\mu ,\alpha \}$ and $\{\alpha ,\mu \}$ .

It is predictive matching of order $O(1/n)$ as shown in . For the ordering of parameters $\{\alpha ,\beta \}$ (so $\alpha ~$ is of main interest), the recommended prior is the one in the last row of the table as it is the reference prior for this ordering.

### Posterior propriety

The marginal posterior for $\alpha$ under the OR $\pi _{1}^{R}(\alpha ,\beta )$ prior and the Jeffreys prior is proper .

### Other properties

Under $\pi _{1}^{R}(\alpha ,\beta )$ the posterior credible interval for $\alpha$ has a frequentist coverage rate of O(n-1) , because it is a Tibshirani prior  .

$\pi _{3}^{R}(\alpha ,\mu )$ can be obtained from $\pi _{1}^{R}(\alpha ,\beta )$ by a change of variable.

Further properties can be found in  and in .

### Alternatives

Jeffreys prior is:

$\pi _{J}(\alpha ,\beta )={\frac {\sqrt {\alpha \psi _{1}(\alpha )-1}}{\beta }}$ 