# compute Dirichlet distribution parameter from known mean distribution

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

• you can achieve the mean-matching you're looking for by setting alpha proportional to mu. You are left with deciding the constant of proportionality, which plays a role similar to the variance parameter of a Gaussian proposal Commented Jul 13, 2022 at 16:40

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.