This document summarizes the Dirichlet Process with Mixed Random Measures (DP-MRM) topic model. DP-MRM is a nonparametric, supervised topic model that does not require specifying the number of topics in advance. It places a Dirichlet process prior over label-specific random measures, with each measure representing the topics for a label. The generative process samples document-topic distributions from these random measures. Inference is done using a Chinese restaurant franchise process. Experiments show DP-MRM can automatically learn label-topic correspondences without manual specification.

1. [Kim+ ICML2012] Dirichlet Process
with Mixed Random Measures : A
Nonparametric Topic Model for
Labeled Data
2012/07/28
Nakatani Shuyo @ Cybozu Labs, Inc
twitter : @shuyo

2. LDA(Latent Dirichlet Allocation)
[Blei+ 03]
• Unsupervised Topic Model
– Each word has an unobserved topic
• Parametric
– The topic size K is given in advance
via Wikipedia

3. Labeled LDA [Ramage+ 09]
• Supervised Topic Model
– Each document has an observed label
• Parametric
via [Ramage+ 09]

4. Generative Process for L-LDA
• 𝜷 𝑘 ~Dir 𝜼
topics corresponding to
𝑑 observed labels
• Λ 𝑘 ~Bernoulli Φ 𝑘
• 𝜽 𝑑 ~Dir 𝜶 𝑑
restricted to labeled
– where 𝜶 𝑑 = 𝛼𝑘 parameters
𝑑
𝑘 Λ 𝑘 =1
𝑑 𝑑
• 𝑧 𝑖 ~Multi 𝜽
𝑑
• 𝑤𝑖 ~Multi 𝜷 𝑧 𝑑
𝑖
via [Ramage+ 09]

5. Pros/Cons of L-LDA
• Pros
– Easy to implement
• Cons via [Ramage+ 09]
– It is necessary to specify label-topic
correspondence manually
• Its performance depends on the corresponds
※) My implementation is here : https://github.com/shuyo/iir/blob/master/lda/llda.py

6. DP-MRM [Kim+ 12]
– Dirichlet Process with Mixed Random Measures
• Supervised Topic Model
• Nonparametric
– K is not the topic size, but the label size
𝛼
𝑁𝑗
𝐻 𝐺0𝑘 𝐺𝑗 𝜃 𝑗𝑖 𝑥 𝑗𝑖
𝜆j 𝑟𝑗 𝐷
𝛽 𝛾𝑘 𝜂
𝐾

7. Generative Process for DP-MRM
𝛼
Each label has a random
measure as topic space 𝑁𝑗
𝐻 𝐺0𝑘 𝐺𝑗 𝜃 𝑗𝑖 𝑥 𝑗𝑖
• 𝐻 = Dir 𝛽
𝜆j 𝑟𝑗 𝐷
• 𝐺0𝑘 ~DP 𝛾 𝑘 , 𝐻 𝛽
𝐾
𝛾𝑘 𝜂
• 𝜆 𝑗 ~Dir 𝒓 𝑗 𝜂 where 𝒓 𝑗 = 𝐼 𝑘∈label 𝑗
• 𝐺 𝑗 ~DP 𝛼, 𝑘∈label 𝑗 𝜆 𝑗𝑘 𝐺0𝑘 mixed random measures
• 𝜃 𝑗𝑖 ~𝐺 𝑗 , 𝑥 𝑗𝑖 ~𝐹 𝜃 𝑗𝑖 = Multi 𝜃 𝑗𝑖

8. Stick Breaking Process
• 𝑣 𝑙 𝑘 ~Beta 1, 𝛾 𝑘 , 𝜋 𝑙𝑘 = 𝑣 𝑙 𝑘 𝑙−1
𝑑=0 1 − 𝑣 𝑑𝑘
• 𝜙 𝑙𝑘 ~𝐻, 𝐺0𝑘 = ∞
𝑙=0 𝜋 𝑙𝑘 𝛿 𝜙 𝑘
𝑙
𝑡−1
• 𝜆 𝑗 ~Dir 𝒓 𝑗 𝜂 , 𝑤 𝑗𝑡 ~Beta 1, 𝛼 , 𝜋 𝑗𝑡 = 𝑤 𝑗𝑡 𝑑=0 1 − 𝑤 𝑗𝑑
𝑘 𝑗𝑡 ∞
• 𝑘 𝑗𝑡 ~Multi 𝜆 𝑗 , 𝜓 𝑗𝑡 ~𝐺0 , 𝐺𝑗 = 𝑡=0 𝜋 𝑗𝑡 𝛿 𝜓 𝑗𝑡

9. Chinese Restaurant Franchise
• 𝑡 𝑗𝑖 : table index of 𝑖-th term in 𝑗-th document
• 𝑘 𝑗𝑡 , 𝑙 𝑗𝑡 : dish indexes on 𝑡-th table of 𝑗-th
document This layer consists on
only a single DP G0
on normal HDP

10. Inference (1)
• Sampling 𝑡

11. Inference (2)
• Sampling 𝑘 and 𝑙

12. Experiments
• DP-MRM gives label-topic probabilistic
corresponding automatically.
via [Kim+ 12]

13. via [Kim+ 12]
• L-LDA can also predict single labeled document to
assign a common second label to any documents.

14. References
• [Kim+ ICML2012] Dirichlet Process with Mixed
Random Measures : A Nonparametric Topic
Model for Labeled Data
• [Ramage+ EMNLP2009] Labeled LDA : A
supervised topic model for credit attribution in
multi-labeled corpora
• [Blei+ 2003] Latent Dirichlet Allocation