# Merits of reparameterizing the Gamma and inverse Gamma

Wikipedia states that the PDFs for the Gamma distribution is:

$$f(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x)$$

However, in Rasmussen 2000, the pdf for the Gamma distribution is specified as: $$p(r) \tilde{} \mathcal{G}(1,\sigma_y^{-2}) \propto r^{-\frac{1}{2}}\exp\Big(-\frac{r}{2\sigma_y^{-2}}\Big)$$ Which means that the pdf used is $$f(x|a,b)\propto x^{\frac{a}{2}-1}\exp\Big(-\frac{x}{2b}\Big)$$

We have the same case for the inverse Gamma distribution, which in Wikipedia is:

$$f(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}\exp\Big(-\frac{\beta}{x}\Big)$$

While Rasmussen 2000 specifies: $$p(x^{-1})~\mathcal{G}(1,1) \implies p(x)\propto x^{-\frac{3}{2}}\exp\Big(-\frac{1}{2x}\Big)$$

Which means that the pdf used is either

$$f(x|a,b)\propto x^{-\frac{a}{2} - 1}\exp\Big(-\frac{1}{2bx}\Big)$$

or

$$f(x|a,b)\propto x^{-\frac{a}{2} - 1}\exp\Big(-\frac{b}{2x}\Big)$$

Because Rasmussen's paper only specifies the PDF up to a normalization constant, the reparameterization of the PDF is acceptable.

However, what is the rationale for specifying the PDF in this manner? Does this make it easier to differentiate with respect to $x$ or the parameters $a,b$, or do properties such as log-concavity arise or something (especially since we're possibly taking $1/b$ instead of $\beta$ in the exponentials)?