I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model with p assigned a beta distribution. The Beta distribution has parameters $\alpha$ and $\beta$ which need a distribution for the fully Bayesian hierarchical model.

In order to create a noninformative distribution he parametrizes the model in terms of $\frac{\alpha}{\alpha+\beta}$ (prior mean) and an approximation of the standard deviation of the beta distribution $(\alpha+\beta)^{-1/2}$ . (described here too http://andrewgelman.com/2009/10/21/some_practical/) On his blog, he mentions not favoring this approach anymore and preferring weakly informative models, but I'd still like to understand the thinking that supports this model.

I have a few questions about this:

  • Why use an approximation here for the parametrization rather than the actual standard deviation of the Beta distribution?
  • How did he arrive at this particular approximation?
  • Bonus: What connection, if any, does this have to a Pareto distribution? I tried parametrizing this model with a Pareto(1.5,1) distribution for $\alpha+\beta$ and a uniform distribution on $\alpha/(\alpha+\beta)$ and ended up with $p(\alpha,\beta)\propto (\alpha+\beta)^{-3/2}$ but Gelman's approach seems to yield $p(\alpha,\beta)\propto (\alpha+\beta)^{-5/2}$ which disagrees with the gentleman writing into Gelman's blog in the link above.

The most detailed explanation of this problem I could find was here http://streaming.stat.iastate.edu/~stat444x/Class%20Notes/6-HierarchicalModels.pdf but this doesn't get at the questions I have.I've been stumped by this for a while and lecture notes available online at a number of universities seem to gloss over why some of these things are done and jump to the joint prior distribution. Any help or resources that could be offered would be really helpful and if this isn't the right forum for this question or I've displayed poor etiquette please tell me (haven't posted before on this).Thank you!

  • $\begingroup$ You should provide enough details to make the question self-contained. $\endgroup$ – Xi'an Oct 18 '18 at 10:54
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    $\begingroup$ The above questions bother me as well, and I was planning to ask myself until I found this post. Would it be possible to reopen it? I personally think it’s clear what is being asked, but I’ll be more than happy to contribute to the description if more details are needed. $\endgroup$ – Ivan Jul 25 '19 at 10:02