Latent Dirichlet Allocation (Blei et al, 2003) is a powerful learning algorithm for automatically and jointly clustering words into "topics" and documents into mixtures of topics. It has been successfully applied to model change in scientific fields over time (Griffiths and Steyvers, 2004; Hall, et al. 2008).
A topic model is, roughly, a hierarchical Bayesian model that associates with each document a probability distribution over "topics", which are in turn distributions over words. For instance, a topic in a collection of newswire might include words about "sports", such as "baseball", "home run", "player", and a document about steroid use in baseball might include "sports", "drugs", and "politics". Note that the labels "sports", "drugs", and "politics", are post-hoc labels assigned by a human, and that the algorithm itself only assigns associate words with probabilities. The task of parameter estimation in these models is to learn both what the topics are, and which documents employ them in what proportions.
Another way to view a topic model is as a generalization of a mixture model like Dirichlet Process Clustering. Starting from a normal mixture model, in which we have a single global mixture of several distributions, we instead say that each document has its own mixture distribution over the globally shared mixture components. Operationally in Dirichlet Process Clustering, each document has its own latent variable drawn from a global mixture that specifies which model it belongs to, while in LDA each word in each document has its own parameter drawn from a document-wide mixture.
The idea is that we use a probabilistic mixture of a number of models that we use to explain some observed data. Each observed data point is assumed to have come from one of the models in the mixture, but we don't know which. The way we deal with that is to use a so-called latent parameter which specifies which model each data point came from.
Collapsed Variational Bayes
The CVB algorithm which is implemented in Mahout for LDA combines advantages of both regular Variational Bayes and Gibbs Sampling. The algorithm relies on modeling dependence of parameters on latest variables which are in turn mutually independent. The algorithm uses 2 methodologies to marginalize out parameters when calculating the joint distribution and the other other is to model the posterior of theta and phi given the inputs z and x.
A common solution to the CVB algorithm is to compute each expectation term by using simple Gaussian approximation which is accurate and requires low computational overhead. The specifics behind the approximation involve computing the sum of the means and variances of the individual Bernoulli variables.
CVB with Gaussian approximation is implemented by tracking the mean and variance and subtracting the mean and variance of the corresponding Bernoulli variables. The computational cost for the algorithm scales on the order of O(K) with each update to q(z(i,j)). Also for each document/word pair only 1 copy of the variational posterior is required over the latent variable.
Invocation and Usage
Mahout's implementation of LDA operates on a collection of SparseVectors of word counts. These word counts should be non-negative integers, though things will-- probably --work fine if you use non-negative reals. (Note that the probabilistic model doesn't make sense if you do!) To create these vectors, it's recommended that you follow the instructions in Creating Vectors From Text, making sure to use TF and not TFIDF as the scorer.
Invocation takes the form:
Topic smoothing should generally be about 50/K, where K is the number of topics. The number of words in the vocabulary can be an upper bound, though it shouldn't be too high (for memory concerns).
Choosing the number of topics is more art than science, and it's recommended that you try several values.
After running LDA you can obtain an output of the computed topics using the LDAPrintTopics utility:
An example is located in mahout/examples/bin/build-reuters.sh. The script automatically downloads the Reuters-21578 corpus, builds a Lucene index and converts the Lucene index to vectors. By uncommenting the last two lines you can then cause it to run LDA on the vectors and finally print the resultant topics to the console.
To adapt the example yourself, you should note that Lucene has specialized support for Reuters, and that building your own index will require some adaptation. The rest should hopefully not differ too much.
We use mean field variational inference to estimate the models. Variational inference can be thought of as a generalization of EM for hierarchical Bayesian models. The E-Step takes the form of, for each document, inferring the posterior probability of each topic for each word in each document. We then take the sufficient statistics and emit them in the form of (log) pseudo-counts for each word in each topic. The M-Step is simply to sum these together and (log) normalize them so that we have a distribution over the entire vocabulary of the corpus for each topic.
In implementation, the E-Step is implemented in the Map, and the M-Step is executed in the reduce step, with the final normalization happening as a post-processing step.