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Let S be the the standard K-1 simplex. Consider the following probability distribution:

$$\begin{align} f(p,\alpha,\beta) &= \prod_{k=1}^K p_k^{\alpha_k-1}(1-p_k)^{\beta_k-1}\\ Z(\alpha,\beta) &= \int_{S} f(p,\alpha,\beta) dp_1dp_2..dp_K\\ \Pr(P=p; \alpha,\beta) &= \frac{f(p,\alpha,\beta) }{Z(\alpha,\beta)} \end{align}$$

How do I draw a sample from this distribution?

Does Z have a closed form?

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  • $\begingroup$ can you give some info on where this comes from? it looks like a multivariate beta distribution, but not quite, so I'm puzzled how it arises. $\endgroup$
    – Carlo Beenakker
    Commented Nov 24, 2017 at 10:36
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    $\begingroup$ You can always use a Metropolis algorithm with proposal moves given by a reflecting random walk on the probability simplex. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Nov 24, 2017 at 11:37
  • $\begingroup$ Gibbs sampling might be a reasonable approach here. $\endgroup$
    – Oliver Nash
    Commented Jan 23, 2018 at 10:45

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