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  • $\begingroup$ Great. And so easy, as you say. Strange that such an important property isn't mentioned in Kotz, Balakrishnan, & Johnson's book. Would you be so kind to add some references about the hyper-Dirichlet and the "very particular prior" to your answer? Cheers! $\endgroup$ Commented Oct 22, 2018 at 8:03
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    $\begingroup$ I think of this as being part of the aggregation property. You can google the Dirichlet-tree prior and get a short paper by Minka that talks about its properties. As far as what Dirichlet prior: notice that, under the construction above, $(f_{00} + f_{10}, f_{01} + f_{11}) \sim \texttt{dirichlet}(A a_{00} + A a_{10}, A a_{01} + A a_{11})$. By the gamma construction, you can also show that these marginal probabilities are independent of the conditional probabilities, so if you choose these particular Dirichlet priors for the marginal and conditionals independently, you get a Dirichlet joint. $\endgroup$ Commented Oct 22, 2018 at 15:35