From beta distribution to Dirichlet: Estimation of the concentrantion parameters Searching at least 3 hours about the connection between beta distribution and dirichlet. My problem is:
I have a collection of random variables $X_i \sim Beta(a_i, b_i)$. The parameters $a_i$ and $b_i$ are known, $\forall i=1,2,...K$. From $\{X_i\}$, I want the dirichlet distribution $(X_1,X_2,...,X_k) \sim Dir(a)$. However, I cannot find a connection between the scale parameters and the concentration vector $a$. The individual distribution of $X_i$ does not provide useful information, but through the dirichlet, the marginals give me the answer. 
Any suggestion?
 A: Beta distribution has $(0, 1)$ support, same as each of the variables jointly distributed as Ditichlet. Given $X_i \sim \mathsf{Beta}(a_i, b_i)$, if you wanted to have something like $(X_1, X_2, \dots, X_k) \sim \mathsf{Dir}(\alpha)$, you would need to force $X_1 + X_2 + \dots + X_k = 1$, because Dirichlet distribution has such constraint. For each draw from the distribution, you would need $x_2 < 1-x_1$, $x_3 < 1 - (x_1 + x_2)$ etc., finally with $x_k = 1 - (x_1 + x_2 + \dots + x_{k-1})$ being deterministic rather then random. This means, that the beta distributions would not be independent any more. What follows, independent beta variables do not jointly follow Dirichlet distribution. 
There is however the reverse relation, if
$$
(Y_1, Y_2, \dots, Y_k) \sim \mathsf{Dir}(\alpha)
$$
then given $\alpha_0 = \sum_{i=1}^k \alpha_i$, marginally $Y_i$'s follow beta distributions
$$
Y_i \sim \mathsf{Beta}(\alpha_i, \alpha_0 - \alpha_i)
$$
So if your variables jointly follow Dirichlet distribution, their marginals are beta distributed. However if the variables are independent and follow beta distributions, then they do not jointly follow Dirichlet distribution, because you wouldn't be able to guarantee the constraint that they sum to unity.
Example
To illustrate this, let's simulate three independent $X_i \sim \mathsf{Beta}(1, 2)$ random variables, and $(Y_1,Y_2,Y_3) \sim \mathsf{Dir}(1, 1, 1)$ variables
library("extraDistr")
n <- 50000
X <- data.frame(V1=rbeta(n, 1, 2), V2=rbeta(n, 1, 2), V3=rbeta(n, 1, 2))
Y <- as.data.frame(rdirichlet(n, c(1, 1, 1)))

If you look at their marginal plots, they all follow the $\mathsf{Beta}(1, 2)$ distribution.

However if you look at the sums of the samples taken from the beta distributions, they clearly do not sum to unity:
range(rowSums(X))
## [1] 0.07868429 2.56876122

Also if you'll compare the scatter plot (with n = 1000 to make it more readable) of the joint distribution of the beta variables, and Dirichlet, you will see that the independent beta variables are less uniformly distributed then Dirichlet.

