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If my likelihood has the form of a beta distribution, and I want to use Jeffreys' prior for its parameters, what is form of the prior?

For some distributions its pretty straight forward to calculate. for example, in the binomial case, the expectation of the second derivative clearly gives you $\operatorname{Beta}(0.5, 0.5)$. But if the likelihood itself has a beta form already, I got lost trying to derive it. Can anybody help me?

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As indicated in this paper by Yang and Berger (1999) that provides a list of Jeffreys priors, the Jeffreys prior associated with the Beta distribution is the determinant of a $2\times 2$ matrix that involves the polygamma function. Nothing close to a standard distribution.

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  • $\begingroup$ Thank you so much. Does that mean it is not possible to sample from the prior then? Also, whats my other option if I want to use an non-informative prior for beta distribution? $\endgroup$ – Babak Mar 13 '17 at 21:20
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    $\begingroup$ You can use this prior as it is computable numerically. $\endgroup$ – Xi'an Mar 14 '17 at 5:22
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    $\begingroup$ Alternatively, using something like $π(α,β)=1/αβ$ would give you a prior easier to handle, although you first need to prove the posterior is then proper. But the same applies to Jeffreys', actually. $\endgroup$ – Xi'an Mar 14 '17 at 8:38

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