## Probabilistic Models

A probabilistic model asserts how observations from a natural phenomenon arise. The model is a joint distribution \begin{aligned} p(\mathbf{x}, \mathbf{z})\end{aligned} of observed variables $$\mathbf{x}$$ corresponding to data, and latent variables $$\mathbf{z}$$ that provide the hidden structure to generate from $$\mathbf{x}$$. The joint distribution factorizes into two components.

The likelihood \begin{aligned} p(\mathbf{x} \mid \mathbf{z})\end{aligned} is a probability distribution that describes how any data $$\mathbf{x}$$ depend on the latent variables $$\mathbf{z}$$. The likelihood posits a data generating process, where the data $$\mathbf{x}$$ are assumed drawn from the likelihood conditioned on a particular hidden pattern described by $$\mathbf{z}$$.

The prior \begin{aligned} p(\mathbf{z})\end{aligned} is a probability distribution that describes the latent variables present in the data. It posits a generating process of the hidden structure.

For details on how to specify a model in Edward, see the model API. We describe several examples in detail in the tutorials.