Parametrization of Dirichlet distribution Hej!
Consider I have a Dirichlet distribution with 4 variables, where the mean (u) values of these are known. $(u1+u2+u3+u4=1)$
Now, I want to obtain the parameters of the Dirichlet distribution ($\alpha_1, \alpha_2, \alpha_3,\alpha_4$) such that the mean in each dimension is retained, while the standard deviation in each dimension is maximized. The distribution needs to be unimodal.
Is there a known solution to this?
Thanks in advance!
 A: According to Wikipedia, if $\alpha_1, \ldots, \alpha_4$ are the parameters, the mean of the $i$th component is $\mu_i = \alpha_i / \sum_{j=1}^4 \alpha_j$ and the variance of the $i$th component is
$$\frac{\mu_i(1-\mu_i)}{1+\sum_{j=1}^4 \alpha_j}.$$
So if the means $\mu_1, \ldots, \mu_4$ are fixed, you can make each component's variances arbitrarily large by choosing $(\alpha_1, \ldots, \alpha_4) = (c\mu_1, \ldots, c\mu_4)$ and take $c \downarrow 0$.

Update: Thank you whuber for pointing out I had forgotten the unimodal requirement. Wikipedia gives the unique mode when $\alpha_i > 1$ for all $i$. If you want to stay in this regime, let $c \downarrow 1/\min_i \mu_i$ so that for any $j$, $\alpha_j  = c \mu_j > \mu_j / \mu_j = 1$.
Unfortunately, you can't set $c = 1/\min_i \mu_i$.
When $\alpha_i = 1$ for some $i$, there are many modes since the density is constant in the $i$th component.
If the minimum of $\mu_1, \ldots, \mu_4$ is unique, then you may set $c=1/\min_i \mu_i$; the distribution will then have a unique mode. (Suppose $\mu_1 < \mu_2 \le \mu_3 \le \mu_4$. Then $\alpha_1 = 1$ while $\alpha_j > 1$ for $j \ne 1$. Maximizing the density will be a concave optimization problem in $x_2, x_3, x_4$ subject to $x_2 + x_3 + x_4 \le 1$, and $x_1$ can then be taken to be $1-(x_2 + x_3 + x_4)$.)
However, if the minimum of the $\mu_1, \ldots, \mu_4$ is achieved by at least two of the $\mu_i$, then setting $c = 1/\min_i \mu_i$ will not work, since we will have $\alpha_i = 1$ for at least two $1$. For instance if $\alpha_1=\alpha_2=1$ and we have some mode $(x_1, x_2, x_3, x_4)$, then $(x'_1, x'_2, x_3, x_4)$ where $x'_1+x'_2 = x_1 + x_2$ is another mode.
When $\alpha_i < 1$ for some $i$, it gets a little hairier. The density tends to infinity as $x_i \to 0$, so there are again many modes.
