A Dirichlet process (DP) can be understood as a distribution over distributions. That is, samples from a Dirichlet process are themselves distributions. A Dirichlet process is characterized by two parameters: a real-valued concentration parameter $ \alpha > 0 $, and a base distribution $ H $. So if we let $ G $ be a random variable distributed according to a Dirichlet process with parameters $ \alpha $ and $ H $, we can express this as:

There are a handful of equivalent representations of Dirichlet processes. One such representation is the Chinese restaurant process. Imagine a restaurant with an infinite number of tables, numbered $ 0, 1, \ldots $ On each table is an independent sample from the base distribution $ H $. When the first customer walks into the restaurant, they sit at table $ 0 $. When the second customer enters, they either sit at table $ 0 $ with probability $ 1/(1 + \alpha) $ or they sit at table $ 1 $ with probability $ \alpha/(1 + \alpha) $. More generally, when the $ n $-th customer enters the restaurant, they choose to sit at a non-empty table $ i $ with probability proportional to the number of people sitting at that table, or they sit at the next empty table with probability proportional to $ \alpha $.

This process is equivalent to the process of drawing samples from $G$.

It is impossible to write a closed-form expression for the distribution $ G $, since its specification requires an infinite amount of information. But for the useful applications of the Dirichlet process, this is not necessary.

A hierarchical Dirichlet process (HDP) is a hierarchy of random variables, where each random variable is distributed according to a Dirichlet process with base distribution given by the parent in the hierarchy. To be more precise, given a tree $ T $. Every node $ \textbf{n} \in T $ is associated with a random variable $ G_{\textbf{n}} $ such that

The HDP allows statistical information to be shared across groups, and is useful for modeling clustered data.

DPs and HDPs are frequently used in mixture models, where the samples from the DP/HDP are not themselves directly observed. Rather, they are inputs to another distribution, which in turn, provides the observed samples. For example, the following is a simple DP mixture model, where the base distribution is Beta and the likelihood is Bernoulli: