The Dirichlet distribution is the conjugate prior to the multinomial sampling distribution. What is the conjugate prior (if any) if the sampling distribution is itself Dirichlet? That is, our samples are nonnegative vectors $\mathbf{p} \in \mathbb{R}^k$ which sum to 1, and we are trying to estimate a distribution over the parameters $\mathbf{\alpha}$ to the Dirichlet distribution that produces $\mathbf{p}$.

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    $\begingroup$ Thanks, that does answer my question: there is no nice conjugate prior. $\endgroup$ Commented May 8, 2015 at 23:26
  • $\begingroup$ Geoffrey, did you ever find out what the conjugate prior of the Dirichlet is? $\endgroup$ Commented Feb 23, 2022 at 0:32
  • $\begingroup$ From that earlier question, I think the answer is that there is no nice answer. $\endgroup$ Commented Feb 24, 2022 at 8:10