How to calculate probability from the Dirichlet distribution

Question

I am trying to understand the Dirichlet distribution, and more specifically, its relationship to the Beta distribution. The way I understand it, it is defined as follows:

where $S_k$ is the probability simplex defined like so:

First of all, what is the rational behind the $B(\alpha)$ term.

Secondly, in MATLAB image we were given below, how do we label the axes? What is each one for? The $\alpha$ values for the MATLAB plot are $\alpha = (0.1, 0.1, 0.1)$

And lastly, how do you calculate probability from this distribution? Is it volume under the curve?

Thanks in advance

Answer 1

Dirichlet distribution is a multivariate generalization of the beta distribution.

1. $B(\alpha)$ is the normalization constant to satisfy the axiom of probability $P(S)=1$.

2. Due to the constraint $S_k$, $x_i = 1 - \sum_{k=1, k\neq i} x_k$. So, the plot seems to me to be of the density function with two of the axis on the lower plane indicating two of $x_i$'s out of three (where the value of third can be derived from those two) and the vertical axis indicate the value of probability density function.

3. For the particular case you showed, the cumulative probability can be calculated by computing the volume under the curve for desired region covered by $x_1$ and $x_2$.