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Exact and Approximate Sum Representations for the Dirichlet Process. Hemnant Ishwaran and Mahmoud Zarepour Presented by: John Paisley. Paper objective.
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Exact and Approximate Sum Representations for the Dirichlet Process Hemnant Ishwaran and Mahmoud Zarepour Presented by: John Paisley
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
Result • I didn’t fully understand how this was arrived at or if there is any important meaning to it.
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)
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.
