## 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 other model tutorials.