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