Latent Dirichlet allocation

In statistics, latent Dirichlet allocation (LDA) is a generative model that allows sets of observations to be explained by unobserved groups which explain why some parts of the data are similar. For example, if observations are words collected into documents, it posits that each document is a mixture of a small number of topics and that each word's creation is attributable to one of the document's topics. LDA was first presented as a graphical model for topic discovery and was developed by David Blei, Andrew Ng, and Michael Jordan in 2003. [cite journal
last = Blei
first = David M.
coauthors = Ng, Andrew Y.; Jordan, Michael I.
title = Latent Dirichlet allocation
journal = Journal of Machine Learning Research
year = 2003
month = January
volume = 3
pages = "pp." 993–1022
url = http://jmlr.csail.mit.edu/papers/v3/blei03a.html
doi = 10.1162/jmlr.2003.3.4-5.993 ]

Topics in LDA

In LDA, each document may be viewed as a mixture of various topics. This is similar to pLSI, except that in LDA the topic distribution is assumed to have a Dirichlet prior. In practice, this results in more reasonable mixtures of topics in a document. It has been noted, however, that the pLSI model is equivalent to the LDA model under a uniform Dirichlet prior distribution. [cite conference
last = Girolami
first = Mark
coauthors = Kaban, A.
title = On an Equivalence between PLSI and LDA
url = http://www.cs.bham.ac.uk/~axk/sigir2003_mgak.pdf
conference = Proceedings of SIGIR 2003
publisher = Association for Computing Machinery
location = New York
year = 2003
id = ISBN 1581136463]

For example, an LDA model might have topics CAT and DOG. The CAT topic has probabilities of generating various words: the words "milk", "meow", "kitten" and of course "cat" will have high probability given this topic. The DOG topic likewise has probabilities of generating each word: "puppy", "bark" and "bone" might have high probability. Words without special relevance, like "the" (see function word), will have roughly even probability between classes (or can be placed into a separate category).

A document is generated by picking a distribution over topics (ie, mostly about DOG, mostly about CAT, or a bit of both), and given this distribution, picking the topic of each specific word. Then words are generated given their topics. (Notice that words are considered to be independent given the topics. This is a standard bag of words model assumption, and makes the individual words exchangeable.)

Model

s.

Inference

Learning the various distributions (the set of topics, their associated word probabilities, the topic of each word, and the particular topic mixture of each document) is a problem of Bayesian inference, which can be carried out using variational methods (or also with Markov Chain Monte Carlo methods, which tend to be quite slow in practice). Alternatively, Thomas Minka and John Lafferty have proposed a method using a variant of expectation propagation. [cite conference
last = Minka
first = Thomas
coauthors = Lafferty, John
title = Expectation-propagation for the generative aspect model
url = https://research.microsoft.com/~minka/papers/aspect/minka-aspect.pdf
conference = Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence
year = 2002
publisher = Morgan Kaufmann
location = San Francisco, CA
id = ISBN 1-55860-897-4]

Applications, extensions and similar techniques

Topic modeling is a classic problem in information retrieval. Related models and techniques are, among others, latent semantic indexing, probabilistic latent semantic indexing, non-negative matrix factorization, Gamma-Poisson.

The LDA model is highly modular and can therefore be easily extended, the main field of interest being the modeling of relations between topics. This is achieved e.g. by using another distribution on the simplex instead of the Dirichlet. The Correlated Topic Model [cite journal
last = Blei
first = David M.
coauthors = Lafferty, John D.
title = Correlated topic models
url = http://www.cs.cmu.edu/~lafferty/pub/ctm.pdf
journal = Advances in Neural Information Processing Systems
volume = 18
year = 2006] follows this approach, inducing a correlation structure between topics by using the logistic normal distribution instead of the Dirichlet. Another extension is the hierarchical LDA (hLDA) [cite conference
last = Blei
first = David M.
coauthors = Jordan, Michael I.; Griffiths, Thomas L.; Tenenbaum; Joshua B
title = Hierarchical Topic Models and the Nested Chinese Restaurant Process
url = http://cocosci.berkeley.edu/tom/papers/ncrp.pdf
conference = Advances in Neural Information Processing Systems 16: Proceedings of the 2003 Conference
publisher = MIT Press
id = ISBN 0262201526
year = 2004] , where topics are joined together in a hierarchy by using the Nested Chinese Restaurant Process.

As noted earlier, PLSA is similar to LDA. The LDA model is essentially the Bayesian version of PLSA model. Bayesian formulation tends to perform better on small datasets because Bayesian methods can avoid overfitting the data. In a very large dataset, the results are probably the same. One difference is that PLSA uses a variable $d$ to represent a document in the training set. So in PLSA, when presented with a document the model hasn't seen before, we fix $Pr(w\; mid\; z)$--the probability of words under topics--to be that learned from the training set and use the same EM algorithm to infer $Pr(z\; mid\; d)$--the topic distribution under $d$. Blei argues that this step is cheating because you are essentially refitting the model to the new data.

External links

* [http://www.cs.princeton.edu/~blei/lda-c/ LDA implementation in C] by Blei.
* [http://chasen.org/~daiti-m/dist/lda/ LDA implementations in C and matlab] .
* [http://gibbslda.sourceforge.net/ LDA implementation in C++ using Gibbs Sampling] .

References and notes