# Basic Question about the Latent Dirichlet Allocation Generative Model

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?

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.)
• $z_n$ follows a $Multinomial(\theta)$ where $\theta$ is the (also latent) topic proportion for the document. $w_n$ is also multinomial, given $z_n$ and $\beta$. – Sean Easter Jun 7 '14 at 3:42