Basic Question about the Latent Dirichlet Allocation Generative Model

Question

So Here is the LDA Generative Model

The $\alpha$ and $\beta$ nodes represent the parameters for two Dirichlet distributions. The $\theta$ and $\phi$ nodes represent the parameters for two multinomial distributions. My question is about the $Z$ node (the topic assignment for the word). It represents a sampled topic from the topic distribution for a document from $\theta$. So I don't think it is actually a random variable, it is a sample. But then why is it a node in the generative model if it is not a random variable? This has me confused. Is $Z$ a random variable or not?

Answer 1

Under the assumed generative model, each topic $z_n$ indexes a distribution over words in the topic, and each $z_n$ is randomly drawn from $Multinomial(\theta)$. Per Blei et al, "The basic idea is that documents are represented as random mixtures over latent topics, where each topic is characterized as a distribution over words."

The keyword is "latent," i.e. unobserved. Under the assumed model, a random topic proportion $\theta$ is chosen for the document. For every word in the document, a random topic index $z_n$ is drawn, and then a random word is drawn "from $p(w_n|z_n, \beta)$, a multinomial probability conditioned on the topic $z_n$." (Blei et al again.)

But all of that is an assumed process under the model, and the only observations are the words in the documents. So even though topics are assumed sampled in the generative process, they're unobserved, random variables.

A great resource is this pair of lectures Blei gave on LDA. It's lengthy but approachable, and he spends a fair bit of time in the first half explaining the assumptions and distributions involved in the generative model.