# Proof of neutrality for dirichlet distribution

I am trying to learn the fields of bayesian non-parametric approaches. I am going thru this manuscript: http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf

I am bit stuck with: Proof of Neutrality for the Dirichlet Basically (page 11), the context is why the stick breaking process generates random vectors from dirichlet distribution.

So, the first challenge for me is to understand the proof? What is the connection of introducing these weird forms of Y

and then how does proving that dirichlet is neutral distribution ties to the proof that stick breaking process generates rv from dirichlet distribution? Thanks

We have $(Q_1,Q_2,\dots,Q_k)\sim\text{Dir}(\alpha)$, and the vector $Q$ sums to one, $\sum_{i=1}^kQ_i=1$.
Now the vector $(Y_1,Y_2,\dots,Y_k)=(\frac{Q_1}{1-Q_k},\frac{Q_2}{1-Q_k},\dots,\frac{Q_{k-1}}{1-Q_k},Q_k)$ is constructed. The first $k-1$ terms of $Y$ sums to one, $\sum_{i=1}^{k-1}Q_i=1$, and the last term is just $Q_k$.
The proof shows that whan $Q\sim\text{Dir}(\alpha)$ the distribution of the vector $Y$ factorizes as a product of a Dirichlet distribution for the fist $k-1$ terms, and a Beta distribution for the last term $Y_k$. Thus we can sample from the Dirichlet distribution by fist simulating $Y_k$ from a Beta distribution, and then simulating the remaining $(Y_1,\dots Y_{k-1})$ from a Dirichlet. To do this, the same approach can be applied again, first simulating the last term from a Beta etc. which is the stick breaking construction.