Statistical genetics: Allele frequencies that follow a Dirichlet distribution

Question

From Foll and Gagiotti (2008) (software BayeScan). They consider a model where several subpopulation are derived from a unique ancestral population.

We consider a set of $I$ loci and let $K_i$ be the number of alleles at the $i^{th}$ locus. The extent of differentiation at locus $i$ between subpopulation $j$ and the ancestral population is measured by $F^{ij}_{ST}$ and is the result of its demographic history. Let $p_i=\{p_{ik}\}$ denote the allele frequencies of the ancestral population at locus $i$, where $p_{ik}$ is the frequency of the allele $k$ at locus $i$ $\left(\sum_k p_{ik} = 1\right)$. We use $\mathbf {p} = \{\mathbf {p_i}\}$ to denote the entire set of allele frequencies of the ancestral population and $\mathbf {\tilde p_{ij}} = \{ \tilde p_{ijk}\}$ to denote the current allele frequencies at locus $i$ for subpopulation $j$. Under these assumptions, the allele frequencies at locus $i$ in subpopulation $j$ follow a Dirichlet distribution with parameters $\theta_{ij}\mathbf {p_i}$,

$$\mathbf {\tilde p_{ij}} \space \tilde \space\space \text{Dir}(\theta_{ij} p_{i1}, ..., \theta_{ij}p_{iK_i})$$

, where

$$\theta_{ij} = \frac{1}{F^{ij}_{ST}}-1$$

^{(I don't have much experience with Dirichlet distributions but I understand its definition and its usefulness in bayesian statistics).}

Can you please help me to understand why $\mathbf {\tilde p_{ij}}$ follows this Dirichlet distribution?

Calling the $j^{th}$ parameter of the Dirichlet distribution, $\alpha_j$, I typically don't understand why they "chose" $\alpha_j = \left(\frac{1}{F^{ij}_{ST}}-1\right) p_{ij}$ and not, say just $\alpha_j = F^{ij}_{ST} p_{ij}$ or anything else.

Answer 1

Largely an informed guess based on properties of distributions rather than specific knowledge of statistical genetics: a beta distribution is useful for modeling the frequencies of two alleles at one locus. A Dirichelet distribution, which is a multivariate generalization of a beta distribution, would thus be useful for modeling a set of loci.

The flip side is to think of beta as just a special case of Dirichelet: if you only have one locus, beta works. If you have multiple loci, then Dirichelet.