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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 are the Dirichlet and the Logit-Normal distributions. Are there any other distributions which come up naturally in this context?

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This is studied in compositional data analysis, there is a book by Aitchison: The Statistical Analysis Of Compositional Data.

Define the simplex by $$ S^n =\{(x_1, \dots,x_{n+1}) \in {\mathbb R}^{n+1} \colon x_1>0,\dots, x_{n+1}>0, \sum_{i=1}^{n+1} x_i=1\}. $$ Note that we use the index $n$ to indicate dimension! Define the geometric mean of an element of the simplex, $x$ as $\tilde{x}$. Then we can define the logratio transformation (introduced by Aitchison) as $x=(x_1, \dots, x_{n+1}) \mapsto (\log(x_1/\tilde{x}), \dots, \log(x_n/\tilde{x})$. This transformation is onto ${\mathbb R}^n$, so have an inverse which I leave to you to calculate (There are also other versions of this transformation that can be used, which has maybe better mathematical properties, more about that later).

Now you can take a normal (or whatever) distribution defined on ${\mathbb R}^n$ and use this inverse transformation to define a distribution on the simplex. The possibilities are limitless, for each and every multivariate distribution on ${\mathbb R}^n$ we get a distribution on the simplex.

I will augment this post later with some examples, and more details on log-ratio transforms.

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