1. Exact and Approximate Sum Representations for the Dirichlet Process
Hemnant Ishwaran and Mahmoud Zarepour
Presented by: John Paisley

2. Paper objective
This paper is concerned with an analytical measure of the closeness of the infinite-dimensional DP to the finite-dimensional DD as seen through the Gamma method for drawing from the DD

3. Result
I didn’t fully understand how this was arrived at or if there is any important meaning to it.

4. Interesting result they mention (but taken from elsewhere)

5. Interesting trick for speeding up VB inference
They represented the DP in this paper in an interesting way.
This is a Dirichlet process after “N” draws. Because $\alpha / K$ goes to zero for the DP, the posterior on the selected components is simply the number of counts. The $\alpha$ on the right represents the weight of all remaining components (which never changes).
We’ve fixed the truncation for VB mixture modeling for theoretical reasons. Also, when using DP, we use stick-breaking to add and subtract component because it is ad-hoc to add and subtract components to a finite DD. The above representation provides a theoretically justifiable way to add and subtract components to the DD. (continued)

6. Continued…
I think we can use the “DD” as on the previous page (actually a DP) in a VB setting. We can subtract unused components with every iteration and not violate any DP rules or be called ad-hoc. I think we can also show that the lower bound guarantee in VB is also not violated.
Why this is good: The stick-breaking prior for DP is a biased prior (Qi presented a way to address this for VB, but it could be called ad-hoc). This prior is symmetric (very important) and is still fully DP. Also, computation time increases linearly as a function of truncation. Therefore, there has been a trade-off: increase truncation for better results, but longer time or vice-versa. Now, because we can theoretically justify pruning with every iteration (IF I’m right), we can literally have both.