compute Dirichlet distribution parameter from known mean distribution

Question

For a particular Bayesian study I am going to apply Dirichlet distribution as my proposal random number generator. I am going to update the distribution parameter every trial based on a given calculated mean of distribution. So, I would like to calculate the parameter, alpha for a multivariate Dirichlet distribution given that I have the mean of marginal distributions. consider a three dimensional Dirichlet distribution with the following means: $mu = (mu_1, mu_2, mu_3)$ given that: $$\mu_i= \frac{a_i}{\sum{a_j}}$$

for a three dimensional distribution, I will have: $$\alpha_1(1-\mu_1)-\mu_1\alpha_2-\mu_1\alpha_3=0$$ $$\alpha_2(1-\mu_2)-\mu_2\alpha_1-\mu_2\alpha_3=0$$ $$\alpha_3(1-\mu_3)-\mu_3\alpha_1-\mu_3\alpha_2=0$$

$$\begin{pmatrix}\alpha_1 \\\ \alpha_2 \\\ \alpha_3\end{pmatrix} . \begin{pmatrix}\ 1-\mu_1 & -\mu_1 & -\mu_1 \\\ 1-\mu_2 & -\mu_2 & -\mu_2 \\\ 1-\mu_3 & -\mu_3 & -\mu_3 \end{pmatrix} = 0$$

Obviously, the coefficients matrix, here the matrix containing $\mu_i$, is singular. Any help to address this problem is highly appreciated.

Thanks Rezgar

Answer 1

As noticed by @JohnMadden in the comment, $\mu_i$ means would be proportional to $\alpha_i$ parameters, so you need to set the parameters appropriately. The parameters control also how anti-concentrated (lower that one), scattered (low) or concentrated (high) the values are, so your choice of scaling factor should reflect how you want them to behave.