I'm reading about Bayesian data analysis by Gelman et al. and I'm having big trouble interpreting the following part in the book (note, the rat tumor rate $\theta $ in the following text has:$\theta \sim Beta(\alpha ,\beta )$

Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution.

Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\alpha ,\beta )$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\alpha}{\alpha +\beta})=\mathrm{log}(\frac{\alpha}{\beta})$ and $\mathrm{log}(\alpha +\beta )$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta $. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on $(-\mathrm{\infty},\mathrm{\infty})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\alpha +\beta )\to \mathrm{\infty}$, and so this particular prior density cannot be used here.

In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,

$p(\alpha ,\beta )\propto (\alpha +\beta {)}^{-5/2},$

and on the natural transformed scale:

$p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\propto \alpha \beta (\alpha +\beta {)}^{-5/2}.$

My problem is especially the bolded parts in the text.

Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$

Question (2): What is the appropriate Jacobian?

Question (3): How does the author arrive into the original and transformed scale priors?

To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.

P.S. if you need more information, or me to clarify my questions please let me know.

Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution.

Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\alpha ,\beta )$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\alpha}{\alpha +\beta})=\mathrm{log}(\frac{\alpha}{\beta})$ and $\mathrm{log}(\alpha +\beta )$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta $. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on $(-\mathrm{\infty},\mathrm{\infty})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\alpha +\beta )\to \mathrm{\infty}$, and so this particular prior density cannot be used here.

In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,

$p(\alpha ,\beta )\propto (\alpha +\beta {)}^{-5/2},$

and on the natural transformed scale:

$p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\propto \alpha \beta (\alpha +\beta {)}^{-5/2}.$

My problem is especially the bolded parts in the text.

Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$

Question (2): What is the appropriate Jacobian?

Question (3): How does the author arrive into the original and transformed scale priors?

To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.

P.S. if you need more information, or me to clarify my questions please let me know.