🚀 go-pugleaf

Hi,
	I am a curious as to what is the common approach for putting a
prior on a Beta distribution.  I know that the  Beta distribution is the
defacto choice for a binomial distribution due to conjugacy .
What I wish to do now is to assume that I am sampling from a Beta
distributon and put some prior on that  Beta distribution. This is a  
Hierarchical Bayes situtation where I wish to represent my uncertainity on
the parameters of the Beta Distribution, say alpha and beta. What is
the best choice of distribution for this meta-prior?

Related question,
	In the Beta distribution we are usually interested in some random 
variable, say X.  Now the likelihood function for X given alpha and beta
is well known.  What is a likelihood function for the alpha and beta 
parameters given X.  Restating this

X ~ Beta(alpha,beta)  
     Likelihood(X |alpha,beta) =
 ((gamma(alpha+beta)/gamma(alpha)*gamma(beta))*X**(alpha-1)*(1-X)**(beta-1)

Ok, then what is 
	Likelihood(alpha,beta | X) = ???????????

I've struggled with this but only succeed in confusing myself.
I appreciate any help I can get on this problem.  Thanks in advance..
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jfmurray@vms2.macc.wisc.edu
jfmurray@WISCMACC.BITNET