Prior for gamma distribution in "mean form"

Question

I need to specify priors for the parameters of a gamma distribution. Normally the gamma distribution is parametrized in either the "rate-form'':

$f\left(x;\alpha,\beta\right)=\frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-\beta x} $

or the "scale-form":

$f\left(x;\kappa,\theta\right)=\frac{1}{\Gamma\left(\kappa\right)\theta^{\kappa}}x^{\kappa-1}e^{-\frac{x}{\theta}}$

However, the only thing I know a priori is something about the mean ( "rate-form": $E\left[x\right]=\frac{\alpha}{\beta}$ or "scale-form" $E\left[x\right]=\kappa\theta$). Is it possible (and legal) to parametrize the gamma distribution as e.g.

$f\left(x;\mu,\beta\right)=\frac{\beta^{\mu\beta}}{\Gamma\left(\mu\beta\right)}x^{\mu\beta-1}e^{-\beta x} $

where $\mu=E\left[x\right]$ and $\beta$ is the rate-parameter from the "rate-form"? Using this form, it would be much easier to specify the priors since I know something about $\mu$.

Btw, I am doing everything simulation-based, so I do not care about getting analytical expressions.

Answer 1

Certainly it's possible to write your gamma in shape-mean form and to place priors on the parameters written in that form.

If we start in 'shape-scale' form:

\begin{eqnarray*} f\left(x;\kappa,\theta\right)&=&\frac{1}{\Gamma\!\left(\kappa\right)\theta^{\kappa}}x^{\kappa-1}e^{-\frac{x}{\theta}}\\ &=&\frac{1}{\Gamma\!\left(\kappa\right)\theta}(\frac{_x}{^\theta})^{\kappa-1}e^{-\frac{x}{\theta}} \end{eqnarray*}

and then write $\mu=\theta\kappa$, equivalently $\theta=\mu/\kappa$, we get:

\begin{eqnarray*} f\left(x;\kappa,\mu\right)&=&\frac{\kappa}{\Gamma\!\left(\kappa\right)\mu}(\frac{_{x\kappa}}{^\mu})^{\kappa-1}e^{-\frac{x\kappa}{\mu}} \end{eqnarray*}

If it helps any, the conjugate prior for $\mu|\kappa$ is inverse-gamma, by inspection.