# Gamma univariate

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

${\displaystyle Ga(x|\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x},\,x>0;\,(\alpha >0,\beta >0).}$

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

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

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

### Formal endorsements

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

It is predictive matching of order ${\displaystyle O(1/n)}$as shown in [1]. For the ordering of parameters ${\displaystyle \{\alpha ,\beta \}}$(so ${\displaystyle \alpha ~}$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 ${\displaystyle \alpha }$ under the OR ${\displaystyle \pi _{1}^{R}(\alpha ,\beta )}$ prior and the Jeffreys prior is proper .[2]

### Other properties

Under ${\displaystyle \pi _{1}^{R}(\alpha ,\beta )}$ the posterior credible interval for ${\displaystyle \alpha }$ has a frequentist coverage rate of O(n-1) [3], because it is a Tibshirani prior [4] .

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

Further properties can be found in [5] and in [1].

### Alternatives

Jeffreys prior is:

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

## References

1. Sun, D. and Ye, K. (1996). Frequentist validity of posterior quantiles for a two-parameter exponential family. Biometrika 83, 55-65.
2. Liseo, B. (1993). Elimination of nuisance parameters with reference priors. Biometrika, 80, 295-304
3. Moala, F. A.  Ramos P. L., Achcar, J.A. (2013). Bayesian inference for two-parameter gamma distribution assuming different noninformative priors. Revista Colombiana de Estadística 36, 321-338.
4. Tibshirani, R. (1989), ‘Noninformative priors for one parameters of many’, Biometrika 76, 604–608.
5. Yang, R. and Berger, J.O. (1996). A catalog of noninformative priors. http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf