I did a bunch of searches on this but did not find anything conclusive. Does the generalized beta distribution of the second kind, also known as the generalized beta prime distribution, have a closed-form conjugate prior? If so, could someone tell me what it is, with a literature citation if you happen to have one handy? If not, I’d like to know that too.
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First, a simple exercise: name all the continuous distributions with support on ${\mathbb R}_+^4$ that you know ... Not so many, right? Now, imagine how likely is that one of them is the conjugate prior of the GB2.
The GB2 distribution has 4 positive parameters. Even if you can obtain a conjugate prior for the 4 parameters of a generalised beta prime distribution, you will translate the problem to "how to sample from that distribution?".
In the case of the beta distribution, which is a submodel of the GB2, the "conjugate prior" is improper. See
http://www.degruyter.com/view/j/spp.2013.4.issue-1/2151-7509.1060/2151-7509.1060.xml
This is a bit discouraging about the more general case GB2.
My suggestion would be to consider reasonable proper priors and to employ an MCMC sampler. There are many already implemented in R, and 4-dimensions is not extremely challenging.
