Computational Statistics – latent Dirichlet allocation

Toy example of LDA

How can we figure out (unsupervised!) the underlying topic of each of a set of documents?¹

Data

Start with documents that each contain words:

Assign topics

As a human, assign topics: Science, Politics, Sports

One potential assignment:

No intuition

What if you don’t have any idea what the words mean (i.e., what if you are the computer)?

Computer assignment of topics

without using the definitions of the words:
Topic 1 , Topic 2, Topic 3

Assign a topic to each word

Assign a topic to each word

Topic frequency

Article property

Property 1: articles are as homogeneous as possible

What about the words?

Word property

Property 2: words are as homogeneous as possible

Goal of LDA:

Color each word with red , green, blue such that

articles are as homogeneous as possible
words are as homogeneous as possible

Gibbs Sampling

Organize the words one at a time, trying to make the articles (Goal #1) and words (Goal #2) as consistent as possible.

$\Rightarrow$ assume that all the other words are correct, and try to predict the topic of a given word.

Predicting the word’s topic

Do we color ball red , green, or blue?

Predicting the word’s topic (Goal #1)

Topic 1	Topic 2	Topic 3
Doc A: how many words in Topic 1	Doc A: how many words in Topic 2	Doc A: how many words in Topic 3
2	0	2

Predicting the word’s topic (Goal #2)

Topic 1	Topic 2	Topic 3
How often is ball in Topic 1	How often is ball in Topic 2	How often is ball in Topic 3
3	1	0

Data without first word

Topic 1	Topic 2	Topic 3
Doc A: how many words in Topic 1	Doc A: how many words in Topic 2	Doc A: how many words in Topic 3
2	0	2
How often is ball in Topic 1	How often is ball in Topic 2	How often is ball in Topic 3
3	1	0

Probability of topic

$\begin{aligned} P (z_{i} = k^{'} | z_{- i}, W) \propto & \frac{# words in d_{i} with k^{'} + α}{# words in d_{i} with any topic + K \cdot α} \cdot \\ \frac{# times i^{t h} word is in topic k^{'} + β}{# words in topic k^{'} + V \cdot β} \end{aligned}$

$z_{i}$ is the topic of the $i^{t h}$ word
$z_{- i}$ are the topics of all the words other than the $i^{t h}$ word
$d_{i}$ is the document of $i^{t h}$ word
$W$ represents all of the words
$K$ is the total number of topics
$V$ is the total number of words
$α$ and $β$ are tuning parameter

Probability that ball is in each topic

$\begin{aligned} P (b a l l_{1} = Topic 1 | z_{- i}, W) \propto & \frac{2 + α}{4 + 3 \cdot α} \cdot \frac{3 + β}{5 + 4 \cdot β} \end{aligned}$

$\begin{aligned} P (b a l l_{1} = Topic 2 | z_{- i}, W) \propto & \frac{0 + α}{4 + 3 \cdot α} \cdot \frac{1 + β}{5 + 4 \cdot β} \end{aligned}$

$\begin{aligned} P (b a l l_{1} = Topic 3 | z_{- i}, W) \propto & \frac{2 + α}{4 + 3 \cdot α} \cdot \frac{0 + β}{5 + 4 \cdot β} \end{aligned}$

Probability that ball is in each topic

$\begin{aligned} P (b a l l_{1} = Topic 1 | z_{- i}, W) \propto & \frac{2 + α \overset{0.3}{}}{4 + 3 \cdot α \overset{0.3}{}} \cdot \frac{3 + β \overset{0.25}{}}{5 + 4 \cdot β \overset{0.25}{}} = 0.2543 \end{aligned}$

$\begin{aligned} P (b a l l_{1} = Topic 2 | z_{- i}, W) \propto & \frac{0 + α \overset{0.3}{}}{4 + 3 \cdot α \overset{0.3}{}} \cdot \frac{1 + β \overset{0.25}{}}{5 + 4 \cdot β \overset{0.25}{}} = 0.0128 \end{aligned}$

$\begin{aligned} P (b a l l_{1} = Topic 3 | z_{- i}, W) \propto & \frac{2 + α \overset{0.3}{}}{4 + 3 \cdot α \overset{0.3}{}} \cdot \frac{0 + β \overset{0.25}{}}{5 + 4 \cdot β \overset{0.25}{}} = 0.0196 \end{aligned}$

$\to$ assign Topic 1 to the first instance of ball

Next step

Update the first instance of ball and move on to the second instance of ball. (Keep iterating!)

What is the topic of a training document?

$Topic (d) = arg max_{k} θ_{d, k}$

where $θ_{d, k}$ is the probability of topic $k$ for document $d$ .

$θ_{d, k}$ is estimated as the proportion of topic $k$ in document $d$ .

What is the topic of a test document?

For each word in the document, you get a probability for each topic, based on the learned topic-word distribution
The probability of a topic $k$ for the new document is essentially the weighted sum of the topic probabilities for all the words in the document, considering their relative importance.

Example (from Math 154)

LDA on TSL

In fall 2015, Benji Lu, Kai Fukutaki, and Ziqi Xiong performed LDA on The Student Life articles for their computational statistics project: https://ziqixiong.shinyapps.io/TopicModeling/

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