# What are some distributions over the probability simplex?

Let $\Delta_{K}$ be the probability simplex of dimension $K-1$, i.e. $x \in \Delta_{K}$ is such that $x_i \ge 0$ and $\sum_i x_i = 1$.

What distributions which are frequently (or well-known, or defined in the past) over $\Delta_{K}$ exist?

Clearly, there is the Dirichlet distribution and the logistic normal distribution. Are there any other distributions which come up naturally in this context?

-