Study notes for Latent Dirichlet Allocation

1. Topic Models

• Topic models are based upon the idea that documents are mixtures of topics, where a topic is a probabilistic distribution over words. A topic model is a generative model for documents: it specifies a simple probabilistic procedure by which documents can be generated (Steyvers and Griffiths, 2007). Two general steps are taken to make a new document: 
  • Step 1, for each document, one chooses a distribution over topics.
  • Step 2, to generate each word in that document, one chooses a topic at random according to the distribution (because one same word may belong to various topics with different probabilities.). Then a word is drawn from the chosen topic in terms of probabilistic sampling, e.g. as illustrated in Figures 1 and 2. 

• When fitting a generative model, the goal is to find the best set of latent variables that can explain the observed data (i.e., observed words in documents), assuming that the model actually generated the data. 
• Many different generative models have been proposed under the same assumption that a document is a mixture of topics, but make slightly different statistical assumptions. 
• The number of topics will affect the interpretability of the results. A solution with too few topics will generally result in very broad topics whereas a solution with too many topics will result in uninterpretable topics that pick out idiosyncratic word combinations. One way is to choose the number of topics that leads to best generalization performance to new tasks. 
• Notations:
  • P(z) is the probability distribution over topics z in a particular document
  • P(w|z) is the probability distribution over words w given topic z. 
  • The model specifies the distribution over words within a document is:
    
    where t is the number of topics, is the probability that the j-th topic is chosen/sampled for the i-th word token, and is the probability of word w_i under topic j.
  • is the multinomial distribution over words for topic j.
  • is the multinomial distribution over topics for document d.
  • D is the number of documents, and each document d consists of N_d words,  is the number of word tokens. 

2. The LDA Model

• Latent Dirichlet Allocation (LDA) is a generative model (i.e., graphic model) that allows sets of observations to be explained by unobserved latent variables that explain why some parts of the data are similar. 
• Different from PLSA, the topic distribution in LDA is assumed to have a Dirichlet prior.
  • Specifically, each document has a Dirichlet prior distribution of topics, and each topic has a Dirichlet prior distribution of words.
  • In practice, this assumption results in more reasonable mixtures of topics in a document. 
  • However, PLSA may be equivalent to the LDA model under a uniform Dirichlet prior distribution.

Toy Example

This exmample is from Edwin Chen's Blog. Suppose you have the following set of sentences: 

• I like to eat broccoli and bananas.
• I ate a banana and spinach smoothie for breakfast.
• Chinchillas and kittens are cute.
• My sister adopted a kitten yesterday.
• Look at this cute hamster munching on a piece of broccoli.

Given these sentences, we look for two topics. LDA might produce something like:

• Sentences 1 and 2: 100% Topic A
• Sentences 3 and 4: 100% Topic B
• Sentence 5: 60% Topic A, 40% Topic B
• Topic A: 30% broccoli, 15% bananas, 10% breakfast, 10% munching, … (at which point, you could interpret topic A to be about food)
• Topic B: 20% chinchillas, 20% kittens, 20% cute, 15% hamster, … (at which point, you could interpret topic B to be about cute animals)

Dirichlet Prior

• Blei et al. (2003) extends the PLSI model by introducing a Dirichlet prior on topics , later Griffiths and Steyvers (2003) enrich it by placing a Dirichlet prior on words . Good choices for the hyperparameters α and β will depend on number of topics and vocabulary size. From previous research, we have found α =50/t and β = 0.01 to work well with many different text collections (Steyvers and Griffiths, 2007).
• Conjugate prior. If the posterior distributions are in the same family as the prior probability distributions, the prior and posterior are then called conjugate distributions, and the prior is called conjugate prior for the likelihood. 

Graphical Model

• The graphical model is represented by plate notation as shown as follows, where the shaded and unshaded variables indicate observed and latent (i.e., unobserved) variables respectively.
  
  where arrows indicate conditional dependencies between variables while plates (the boxes in the figure) refer to repetitions of sampling steps with the variable in the lower right corner referring to the number of samples. For example, the inner plate over z and w illustrates the repeated sampling of topics and words until N_d words have been generated for document d.
• The topic models can be interpreted by matrix factorization, illustrated as follows together with LSA interpretations. The word-document co-occurrence matrix is split into two matrices: a topic matrix and a document matrix.Note that the constraints in LDA are that the feature values (topic distributions) are non-negative and should be summed up to one.  LSA decomposition (i.e., SVD factorization) does not have such constraints. 
  

Gibbs Sampling

• The challenge is to efficiently estimate the posterior distributions and, according to the large number of word tokens in the document collections. 
• Gibbs sampling (a.k.a alternating conditional sampling) is a specific form of Markov chain Monte Carlo, simulating a high-dimensional distribution by sampling on lower-dimensional subsets of variables where each subset is conditioned on the value of all others. The sampling is done sequentially and proceeds until the sampled values approximate the target distribution.
• Markov chain Monte Carlo (MCMC) refers to a set of approximate iterative techniques designed to sample values from complex (often high-dimensional) distributions. More lectures about MCMC can be referred to here.
• The procedure is as follows (assuming K topics): 
  • Go through each document, and randomly assign each word in the document to one of the K topics.
  • Notice that this random assignment already  gives you both topic representations of all the documents, and word distributions of all the topics (albeit not very good ones). 
  • So to improve on them, for each document d ...
    • Go through each word w in d, and for each topic t, compute two things: 
      • p(t|d) = the proportion of words in document d that are currently assigned to topic t.
      • p(w|t) = the proportion of assignments to topic t over all documents that come from this word w. 
      • we compute that p(w|d)=p(w|t)p(t|d). 
  • After repeating the previous step a large number of times, you’ll eventually reach a roughly steady state where your assignments are pretty good. So use these assignments to estimate the topic mixtures of each document (by counting the proportion of words assigned to each topic within that document) and the words associated to each topic (by counting the proportion of words assigned to each topic overall).

Generative Process

• Decide the number of words N that a document will have, according to a Poisson distribution. 
• Draw a topic distribution, , where is a draw from a uniform Dirichlet distribution with scaling parameter .
• For each word in the document
  • Draw a specific topic , where is a multinomial distribution.
  • Draw a word  using the picked topic. 
  Assuming this generative model for a collection of documents, LDA then tries to backtrack from the documents to find a set of topics that are likely to have generated the collection. 

Example. To generate some particular document D, you might:

• Pick 5 to be the number of words in D.
• Decide that D will be 1/2 about food and 1/2 about cute animals.
• Pick the first word to come from the food topic, which then gives you the word “broccoli”.
• Pick the second word to come from the cute animals topic, which gives you “panda”.
• Pick the third word to come from the cute animals topic, giving you “adorable”.
• Pick the fourth word to come from the food topic, giving you “cherries”.
• Pick the fifth word to come from the food topic, giving you “eating”.So the document generated under the LDA model will be “broccoli panda adorable cherries eating” (note that LDA is a bag-of-words model).

3. Computing Similarity

• The derived topic probability distributions can be used to compute document or word similarity.
• Two documents are similar to the extent that the same topics appear in these documents.
• Two words are similar to the extent that they appear in the same topic. 
• My understanding: if we look at the matrix representation, it is easy to compute both document and word similarities according to topic distributions. 

Document Similarity

• Given the topic distributions ,  of two documents, the document similarity is measured as that of the topic distributions. 
• A standard function to measure the difference or divergence between two distributions p and q is the Kullback Leibler (KL) divergence:
  
• The KL divergence is asymmetric and in many applications, it is convenient to apply a symmetric measure based on KL divergence:
  
• Another option is to apply the symmetrized Jensen-Shannon (JS) divergence:
  
• If we treat the topic distributions as a vector, then other measures such as Euclidian distance, cosine similarity can be applied. 

Word Similarity

• The word similarity can be measured by the extent that they share the same topics. Specifically, one word can appear in many topics. Hence the word similarity can be regarded as their overlappings. 
• Given two word-topic distributions  and , either the symmetrized KL or JS divergence can be used to compute word similarity. 

References

1. Edwin Chen, Introduction to Latent Dirichlet Allocation.
2. Blei et al., 2003, Latent Dirichlet Allocation, Journal of Machine Learning Research.
3. Griffiths and Steyvers, 2003, Prediction and semantic association, In Neural information processing systems.
   
4. Steyvers and Griffiths, 2007, Probabilistic Topic Models, Handbook of latent semantic analysis.