Michael Data

Binary and Count Data is a d-dimensional vector, where corresponds to the conditional probability of given .

Typically, the data matrix will be sparse.

A mixture model for such data: where are the parameters of the k-th component.

Mixture of Multinomials  Each component is a multinomial, like a die with d sides.

Generative Model

• * For each document ,
• * n_i is the total word count for the document (assume it's known)
• * sample k from • * for r = 1 to n_i,
• sample from • * end
• * = # of times word j was sampled

Mixture of Conditional Independence Models

For simplicity, assume are binary. , but now and they do not sum to 1.
Now each word is a conditionally independent binary random variable.

Generative Model

• * For each document ,
• * sample k from • * for j = 1 to d,
• is sampled from • * end

Latent Dirichlet Allocation Model

[http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation Latent Dirichlet Allocation], LDA, or Topic Model: Model that allows each word to belong to a different multinomial component.

EM doesn't work very well for this. The major difference is that Z is now at the word level. Allows for documents to be mixtures of topics.

Learning the Parameters

Use the Expectation Maximization algorithm.

1. In the E-step, use Bayes' rule to compute membership probabilities

#*

1. In the M-step, maximize the log-likelihood of the data

#* #* For multinomial model, #
#* For conditional independence model, $\phi_i^{new} = \frac{1}{N_k} \sum_{i = 1}^N w_{ik} x_{ij}$
# acts as an indicator variable 