 rigliztetbf

2022-06-26

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:$\theta \sim Beta\left(\alpha ,\beta \right)$
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 $\left(\alpha ,\beta \right)$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}\left(\frac{\alpha }{\alpha +\beta }\right)=\mathrm{log}\left(\frac{\alpha }{\beta }\right)$ and $\mathrm{log}\left(\alpha +\beta \right)$, 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 $\left(-\mathrm{\infty },\mathrm{\infty }\right)$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $\left(\alpha +\beta \right)\to \mathrm{\infty }$, 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 $\left(\frac{\alpha }{\alpha +\beta },\left(\alpha +\beta {\right)}^{-1/2}\right)$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,
$p\left(\alpha ,\beta \right)\propto \left(\alpha +\beta {\right)}^{-5/2},$
and on the natural transformed scale:
$p\left(\mathrm{log}\left(\frac{\alpha }{\beta }\right),\mathrm{log}\left(\alpha +\beta \right)\right)\propto \alpha \beta \left(\alpha +\beta {\right)}^{-5/2}.$
My problem is especially the bolded parts in the text.
Question (1): What does the author explicitly mean by: "is uniform on $\left(\frac{\alpha }{\alpha +\beta },\left(\alpha +\beta {\right)}^{-1/2}\right)$
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. Raven Higgins

If anyone runs into the similar section in Gelman's book, I'm going to give my own solution that I came up with below (pages 110-111).
By this, the author only implies that

When the author refers to a "appropriate Jacobian," he is referring to the Jacobian matrix's determinant in the formula for density functions with changed variables:

Simply said, the author uses the change of variables formula twice. The fact that

If we denote and ), then:

From the formula for variable change, we obtain:

and there it is, i.e. the prior in original scale.
By using the exact same modification of variables for the alternate scale:

where this time and  We obtain: for the Jacobian determinant.

so we get:

or

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