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 . Before assigning a hyperprior distribution, we reparameterize in terms of and , which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for . 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 scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit , 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 , which when multiplied by the appropriate Jacobian yields the following densities on the original scale,
and on the natural transformed scale:
My problem is especially the bolded parts in the text.
Question (1): What does the author explicitly mean by: "is uniform on
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.