Latent Dirichlet Allocation Basics


Weiyang Wang
Undergraduate Researcher

July 22, 2023

1. Introduction

This article explores the fundamental aspects of Latent Dirichlet Allocation (LDA), a highly-utilized unsupervised probabilistic technique for topic modeling. It elaborates the core principles of LDA, providing an accessible interpretation of the underlying mathematical concepts that dictate the operation of the model, as well as the training process using Gibbs Sampling with operable illustration. The real-world applications and potential extensions of the LDA model are also explored.

2. Model Assumption

3. Model Intuition

As stated in section 2, The fundamental idea behind LDA is that every document in the text corpus is a mixture of topics, and each word in a document is attributable to one of these topics.
By saying a document is "generated" by its given topic distribution, it means that each document is assumed to be produced in the following manner:

In this way, the document is "generated" by its given topic distribution, and each word in turn is generated by the topic distribution of its respective document and the word distribution of its assigned topic.
The task of LDA is to reverse this process. Given a corpus of documents, LDA tries to figure out the topic distributions for each document and the word distributions for each topic that would have most likely resulted in the observed corpus.

4. Theorem Explaination

4.1 Formula

\(P(\boldsymbol{W},\boldsymbol{Z},\theta,\varphi;\alpha,\beta) = \prod_{j=1}^{M}P(\theta_j;\alpha) \prod_{i=1}^{K}P(\varphi_i;\beta) \prod_{t=1}^{N}P(Z_{j,t} \mid \theta_j) P(W_{j,t} \mid \varphi_{Z_{j,t}})\)

4.2 Explanation

The formula represents the joint probability distribution for a LDA model, denoted as \(P(\boldsymbol{W},\boldsymbol{Z},\theta,\varphi;\alpha,\beta)\). Here's what each part means:

\(P(\boldsymbol{W},\boldsymbol{Z},\theta,\varphi;\alpha,\beta)\) is the joint probability of the observed words \(\boldsymbol{W}\), the latent (or hidden) topic assignments \(\boldsymbol{Z}\), the per-document topic proportions \(\theta\), and the per-topic word probabilities \(\varphi\), given the Dirichlet prior parameters \(\alpha\) and \(\beta\).

\(\prod_{j=1}^{M}P(\theta_j;\alpha)\) is the probability of the topic distribution for each document \(j\) under a Dirichlet prior \(\alpha\).

\(\prod_{i=1}^{K}P(\varphi_i;\beta)\) is the probability of the word distribution for each topic \(i\) under a Dirichlet prior \(\beta\).

\(\prod_{t=1}^{N}P(Z_{j,t} \mid \theta_j)\) is the probability of the topic assignments \(Z_{j,t}\) for each word \(t\) in each document \(j\), given the topic distribution \(\theta_j\) of that document.

\(P(W_{j,t} \mid \varphi_{Z_{j,t}})\) is the probability of each word \(W_{j,t}\) in each document \(j\), given the word distribution \(\varphi_{Z_{j,t}}\) of the assigned topic \(Z_{j,t}\) for that word.

The aim of the LDA model is to find values for \(\boldsymbol{Z}\), \(\theta\), and \(\varphi\) that maximize this joint probability, given the observed words \(\boldsymbol{W}\) and the priors \(\alpha\) and \(\beta\). Due to the complexity of this problem, approximation methods like Gibbs sampling or variational inference are often used to estimate these values.

5. Training - Gibbs Sampling

5.1 Objective

The principal aim of Gibbs Sampling in LDA is to maximize the monochromaticity of topic distribution within documents, and word distribution within topics. Implicitly, this implies a desirable model where each document is characterized by one or a limited number of topics, and similarly, each word belongs to a limited number of topics.

5.2 Procedure

The following steps outline the typical steps for training:

5.3 Hyperparameters

The model depends on several critical hyperparameters:

5.4 Initialization of Dirichlet Priors

There are several strategies for initializing the Dirichlet priors:

5.5 Animation

6. Basic Applications of LDA

6.1 Topic Modeling

The basic implementation of LDA, as stated above, takes a set of documents as input, and is able to give the topic distribution for each document. However, LDA doesn't output nominal topics directly, which means that the topics within the result don't have any nominal meanings themselves. By studying the words that compose each topic, it is possible to assign names and labels to the topics, but this requires further steps and techniques.

6.2 Document Clustering

LDA can be used to cluster large collections of text documents into topics, which can help with tasks such as information retrieval, document classification, and recommendation systems.

6.3 Content-based recommendation

LDA can be used to identify the topics that a user is interested in based on their past behavior or preferences, and then recommend similar content that matches those topics.

6.4 Sentiment Analysis

LDA can be used to analyze the sentiment of a piece of text by identifying the topics that are most commonly associated with positive or negative sentiment.

7. Extended Applications of LDA

As described so far, LDA is based on the distribution of words in a corpus of text, often it is directly applied to text objects, but it can also handle other problems with proper feature transformations.

7.1 Fraud Detection

7.2 Medical diagnosis

8. Additional Resources