# Prior of multivariate Polya distribution?

Anyone knows a prior (preferably conjugate) to the multivariate Polya distribution?

I need it for Gibbs sampling. So if anyone has another idea, I am interested.

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You could choose the gamma distribution for the parameters $\alpha_i$ of the polya distribution. Since, you are doing a bayesian analysis you could sample the intermediate probabilities $p$ instead of working with the pdf $\text{Pr}(X|\alpha)$.

In other words, the multivariate polya is given by:

$\text{Pr}(X|\alpha) = \int_p(\ \text{Pr}(X|p) \ \text{Pr}(p|\alpha) \ ) \ dp$

In a bayesian analysis you do not have to actually compute the above integral but actually work with $\text{Pr}(X|p)$ and $\text{Pr}(p|\alpha)$ directly.

$\text{Pr}(X|p)$ which gives the probability model for the data,

$\text{Pr}(p|\alpha)$ which gives the prior for the probability vector $p$

and

$f(\alpha|-)$ is the hyperprior for $\alpha$.

So my suggestion amounts to selecting a gamma distribution as the hyperprior for the individual components of $\alpha$.

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The slight glitch that may make peri4n unhappy is that the Gamma distribution is not conjugate in this case, because of the Gamma functions in the Dirichlet density $\text{Pr}(p|\alpha)$. –  Xi'an Nov 27 '11 at 7:06
@Xi'an am not aware of a conjugate prior for for the dirichlet density. However, a google search turned up this post at MO which may be useful: Conjugate prior of the Dirichlet distribution? –  varty Nov 27 '11 at 13:50
there is a formal conjugate to the Dirichlet distribution because it is an exponential family. See, e.g., Chapter 3 of my book The Bayesian Choice but in the case of the Dirichlet is is a non-standard distribution $$\pi(\alpha_1,\ldots,\alpha_p) \propto \dfrac{\Gamma(\sum_i \alpha_i)^{n_0}}{\prod_i \Gamma(\alpha_i)^{n_0}} \prod_i \delta_i^{\alpha_i} \qquad n_0>0\,,\ 0<\delta_i<1.$$ –  Xi'an Nov 28 '11 at 7:08