# 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)?

## 1 Answer

Rasmussen selects a vague prior by the provided selection of parameters, see the following quote from the paper you mentioned:

The shape parameter of the Gamma prior is set to unity, corresponding to a very broad (vague) distribution

Similar thing happens to Inverse Gamma prior.

The reason behind selection of a vague prior is to make the prior less informative and reduce effects of the prior on the posterior. It is called Objective Bayesian approach to the prior selection.

P.S. There is a theory behind selection of the most noniformative prior e.g. reference and Jeffreys priors try to reduce ammount of information in the prior.

• Thank you for the answer. However, my question is not why he picks the vague priors, but why does he use a different parameterization of said priors (why is the formula of the PDF different?) – peco Aug 3 '17 at 12:20