Graphical Models

• Nodes are random variables
• Edges denote possible dependence
• Observed variables are shaded
• Plates denote replicated structure
• Structure of the graph defines the pattern of conditional dependence between the ensemble of random variables
• E.g., this graph corresponds to

$$p(y, x_{1}, ..., x_{N}) = p(y)\prod^{N}_{n=1}p(x_{n}|y)$$

Generative Model

• Each document is a random mixture of corpus-wide topics
• Each word is drawn from one of those topics

``` ```

``` ```

The Posterior Distribution

• In reality, we only observe the documents
• LDA is to infer the underlying topic structure

``` ```

Latent Dirichlet allocation

``` ```

where, $\beta_{k}$ ~ $Dir(\eta)$, $\theta_{d}$ ~ $Dir(\alpha)$, $z_{d,n}$ ~ $Multinomial(\theta_{d})$ , $w_{d,n}$ ~ $Multinomial(\beta_{z_{d,n}})$