1. Latent Dirichlet Analysis
CS246
Latent Dirichlet Analysis

2. LSI LSI uses SVD to find the best rank-K approximation
The result is difficult to interpret especially with negative numbers
Q: Can we develop a more interpretable method?

3. Theory of LDA (Model-based Approach)
Develop a simplified model on how users write a document based on topics.
Fit the model to the existing corpus and “reverse engineer” the topics used in a document
Q: How do we write a document?
A: (1) Pick the topic(s) (2) Start writing on the topic(s) with related terms

4. Two Probability Vectors
For every document d, we assume that the user will first pick the topics to write about
P(z|d) : probability to pick topic z when the user write each word in document d.
Document-topic vector of d
We also assume that every topic is associated with each term with certain probability
P(w|z) : the probability of picking the term w when the user write on the topic z.
Topic-term vector of z

5. Probabilistic Topic Model
There exists T number of topics
The topics-term vector for each topic is set before any document is written
P(wj|zi) is set for every zi and wj
Then for every document d,
The user decides the topics to write on, i.e., P(zi|d)
For each word in d
The user selects a topic zi with probability P(zi|d)
The user selects a word wj with probability P(wj|zi)

6. Probabilistic Document Model
P(w|z)
P(z|d)
Topic 1
Topic 2
1.0
money1 bank1 loan1 bank1 money1 ...
bank
loan
money
DOC 1
0.5
money1 river2 bank1 stream2 bank2 ...
DOC 2
river
stream
bank
1.0
river2 stream2 river2 bank2 stream2 ...
DOC 3

7. Example: Calculating Probability
z1 = {w1:0.8, w2:0.1, w3:0.1} z2 = {w1:0.1, w2:0.2, w3:0.7}
d’s topics are {z1: 0.9, z2:0.1} d has three terms {w32, w11, w21}.
Q: What is the probability that a user will write such a document?

8. Corpus Generation Probability
T: # topics
D: # documents
M: # words per document
Probability of generating the corpus C

9. Generative Model vs Inference (1)
P(w|z)
P(z|d)
Topic 1
Topic 2
1.0
money1 bank1 loan1 bank1 money1 ...
bank
loan
money
DOC 1
0.5
money1 river2 bank1 stream2 bank2 ...
DOC 2
river
stream
bank
1.0
river2 stream2 river2 bank2 stream2 ...
DOC 3

10. Generative Model vs Inference (2)
Topic 1
Topic 2 ?
money? bank? loan? bank? money? ... ?
DOC 1 ?
money? river? bank? stream? bank? ...
DOC 2 ? ?
river? stream? river? bank? stream? ...
DOC 3

11. Probabilistic Latent Semantic Index (pLSI)
Basic Idea: We pick P(zj|di), P(wk|zj), and zij values to maximize the corpus generation probability
Maximum-likelihood estimation (MLE)
More discussion later on how to compute the P(zj|di), P(wk|zj), and zij values that maximize the probability

12. Problem of pLSI
Q: 1M documents, 1000 topics, 1M words words/doc. How much input data? How many variables do we have to estimate?
Q: Too much freedom. How can we avoid overfitting problem?
A: Adding constraints to reduce degree of freedom

13. Latent Dirichlet Analysis (LDA)
When term probabilities are selected for each topic
Topic-term probability vector, (P(w1|zj), …, P(wW|zj)), is sampled randomly from Dirichlet distribution
When users select topics for a document
Document-topic probability vector, (P(z1|d), …, P(zT|d)), is sampled randomly from Dirichlet distribution

14. What is Dirichlet Distribution?
Multinomial distribution
Given the probability pi of each event ei, what is the probability that each event ei occurs ⍺i times after n trial?
We assume pi’s. The distribution assigns ⍺i’s probability.
Dirichlet distribution
“Inverse” of multinomial distribution: We assume ⍺i’s. The distribution assigns pi’s probability.

15. Dirichlet Distribution
Q: Given ⍺1, ⍺2,…, ⍺k, what are the most likely p1, p2, pk values?

16. Normalized Probability Vector and Simplex
Remember that and
When (p1, …, pn) satisfies p1 + … + pn = 1, they are on a “simplex plane”
(p1, p2, p3) and their 2-simplex plane

17. Effect of ⍺ values p1 p2 p3 p1 p2 p3

18. Effect of ⍺ values p1 p2 p3 p1 p2 p3

19. Effect of ⍺ values p1 p2 p3 p1 p2 p3

20. Effect of ⍺ values p1 p1 p3 p3 p2 p2

21. Minor Correction is not “standard” Dirichlet distribution.
The “standard” Dirichlet Distribution formula:
Used non-standard to make the connection to multinomial distribution clear
From now on, we use the standard formula

22. Back to LDA Document Generation Model
For each topic z
Pick the word probability vector P(wj|z)’s by taking a random sample from Dir(β1,…, βW)
For every document d
The user decides its topic vector P(zi|d)’s by taking a random sample from Dir(⍺1,…, ⍺T)
For each word in d
The user selects a topic z with probability P(z|d)
The user selects a word w with probability P(w|z)
Once all is said and done, we have
P(wj|z): topic-term vector for each topic
P(zi|d): document-topic vector for each document
Topic assignment to every word in each document

23. Symmetric Dirichlet Distribution
In principle, we need to assume two vectors, (⍺1,…, ⍺T) and (β1 ,…, βW) as input parameters.
In practice, we often assume all ⍺i’s are equal to ⍺ and all βi’s = β
Use two scalar values ⍺ and β, not two vectors.
Symmetric Dirichlet distribution
Q: What is the implication of this assumption?

24. Effect of ⍺ value on Symmetric Dirichlet
Q: What does it mean? How will the sampled document topic vectors change as ⍺ grows?
Common choice: ⍺ = 50/T, b = 200/W p1 p1 p3 p3 p2 p2

25. Plate Notation a
P(z|d) z b w
P(w|z) M T N

26. LDA as Topic Inference
Given a corpus d1: w11, w12, …, w1m … dN: wN1, wN2, …, wNm
Find P(z|d), P(w|z), zij that are most “consistent” with the given corpus
Q: How can we compute such P(z|d), P(w|z), zij?
The best method so far is to use Monte Carlo method together with Gibbs sampling

27. Monte Carlo Method (1)
Class of methods that compute a number through repeated random sampling of certain event(s).
Q: How can we compute Pi?

28. Monte Carlo Method (2) Define the domain of possible events
Generate the events randomly from the domain using a certain probability distribution
Perform a deterministic computation using the events
Aggregate the results of the individual computation into the final result
Q: How can we take random samples from a particular distribution?

29. Gibbs Sampling
Q: How can we take a random sample x from the distribution f(x)?
Q: How can we take a random sample (x, y) from the distribution f(x, y)?
Gibbs sampling
Given current sample (x1, …, xn), pick an axis xi, and take a random sample of xi value assuming all other (x1, …, xn) values
In practice, we iterative over xi’s sequentially

30. Markov-Chain Monte-Carlo Method (MCMC)
Gibbs sampling is in the class of Markov Chain sampling
Next sample depends only on the current sample
Markov-Chain Monte-Carlo Method
Generate random events using Markov-Chain sampling and apply Monte-Carlo method to compute the result

31. Applying MCMC to LDA
Let us apply Monte Carlo method to estimate LDA parameters.
Q: How can we map the LDA inference problem to random events?
We first focus on identifying topics {zij} for each word {wij}.
Event: Assignment of the topics {zij} to wij’s. The assignment should be done according to P({zij}|C)
Q: How to sample according to P({zij}|C)?
Q: Can we use Gibbs sampling? How will it work?
Q: What is P(zij|{z-ij},C)?

32. nwt: how many times the word w has been assigned to the topic t
ndt: how many words in the document d have been assigned to the topic t
Q: What is the meaning of each term?

33. LDA with Gibbs Sampling
For each word wij
Assign to topic t with probability
For the prior topic t of wij, decrease nwt and ndt by 1
For the new topic t of wij, increase nwt and ndt by 1
Repeat the process many times
At least hundreds of times
Once the process is over, we have
zij for every wij
nwt and ndt

34. Result of LDA (Latent Dirichlet Analysis)
TASA corpus
37,000 text passages from educational materials collected by Touchstone Applied Science Associates
Set T=300 (300 topics)

35. Inferred Topics

36. Word Topic Assignments

37. LDA Algorithm: Simulation
Two topics: River, Money Five words: “river”, “stream”, “bank”, “money”, “loan”
Generate 16 documents by randomly mixing the two topics and using the LDA model
river
stream
bank
money
loan
River
1/3
Money

38. Generated Documents and Initial Topic Assignment before Inference
First 6 and the last 3 documents are purely from one topic. Others are mixture White dot: “River”. Black dot: “Money”

39. Topic Assignment After LDA Inference
First 6 and the last 3 documents are purely from one topic. Others are mixture After 64 iterations

40. Inferred Topic-Term Matrix
Model parameter
Estimated parameter
Not perfect, but very close especially given the small data size
river
stream
bank
money
loan
River
0.33
Money
river
stream
bank
money
loan
River
0.25
0.4
0.35
Money
0.32
0.29
0.39

41. X = SVD vs LDA Both perform the following decomposition
SVD views this as matrix approximation
LDA views this as probabilistic inference based on a generative model
Each entry corresponds to “probability”: better interpretability
topic
term
term
topic X =
doc
doc

42. LDA as Soft Classfication
Soft vs hard clustering/classification
After LDA, every document is assigned to a small number of topics with some weights
Documents are not assigned exclusively to a topic
Soft clustering

43. Summary Probabilistic Topic Model Statistical parameter estimation
Generative model of documents
pLSI and overfitting
LDA, MCMC, and probabilistic interpretation
Statistical parameter estimation
Multinomial distribution and Dirichlet distribution
Monte Carlo method
Gibbs sampling
Markov-Chain class of sampling