Stick-breaking construction of Dirichlet process In the stick-breaking construction of Dirichlet (let me base things on Sethuraman's construction - slide 6 of this) do we sample one $\phi$ vector from the base distribution $H$ and use it for sampling $\phi_k$ at each step $k$? This is what I "think" they actually do. 
But I am puzzled, as in their paper they mention $H$ is a symmetric Dirichlet distribution over the vocabulary (see the third paragraph of section 2 of this). Which means the dimensionality of $\phi$ would be the size of the vocabulary. Then it is not clear how the index $k$ of $\phi_k$s are mapped to the index of $\beta_k$s. Basically sampling $\beta_k$ will stop if no more stick is left, which means the $k$ index of $\beta_k$ can potentially be much smaller than the size of the $\phi$ vector of size $|\text{Vocabulary}|$ sampled from $H$.
My guess is that they first sample $\phi_k$s, but all they keep from the sample is its corresponding index in $\phi$. Then the sampled $\beta_k$ is just a weight associated with that index. This ways it is clear what each $\beta_k$ corresponds to.
 A: From what I understand here, they are explaining how to build a Dirichlet process from a base distribution H. 
It means that each $\phi_k$ is sampled from H directly. Each $\phi_k$ corresponds to a topic. It is each of these $\phi_k$ which are the size of the vocabulary.
Example : if your vocabulary is of size $N$, each $\phi_k$ can be represented as a vector from $[0, 1]^N$ which corresponds to the probability of each word of appearing in a document of this topic.
The weights are determined through a $GEM(\gamma)$ : even though I'm not sure how I clearly understood how it works, it seems to sequentially assign a weight on each distribution $\phi_k$ until the sum of the weigths is 1, when it assigns then a 0 weight for all subsequent $\phi_k$s
Basically, your $k$ s correspond to the index of the topics, and the $\beta_k$s are drawn indepentendly from the $\phi_k$s.
Don't hesitate to tell me if something is not clear, most of what I tried to explain comes from a paper I worked on last semester on a related subject (Indian Buffet Process)
