---
tag: 7174H28FT6
kind: definition
name: Dirichlet process
msc: [60G57, 62G05, 62F15]
defines: XT4SWG6J7N
uses: [0VGMEBVW1M, 04WSW8T3SY, RX4CVSSRRJ, 099SXAG014, 0EW824Z7G2]
sources:
  - url: https://en.wikipedia.org/wiki/Dirichlet_process
---

## Definition

Let $(X, \mathcal{A})$ be a measurable space, where $\mathcal{A}$ is a [[sigma-algebra|04WSW8T3SY]] on $X$.
Let $H$ be a [[probability measure|0VGMEBVW1M]] on $(X, \mathcal{A})$ and let $\alpha > 0$ be a real number.
A random probability measure $G$ on $(X, \mathcal{A})$ is a *Dirichlet process* with concentration parameter $\alpha$ and base measure $H$ if the following holds.
For every finite partition $A_1, \ldots, A_r$ of $X$ into sets of $\mathcal{A}$, the random vector $(G(A_1), \ldots, G(A_r))$ has the [[Dirichlet distribution|RX4CVSSRRJ]] with parameters $(\alpha H(A_1), \ldots, \alpha H(A_r))$.
Here a parameter equal to $0$ means that the corresponding coordinate is $0$ almost surely.
One writes $G \sim \mathrm{DP}(\alpha, H)$.

## Remarks

Example: take $A \in \mathcal{A}$ with $0 < H(A) < 1$, and the partition $A, X \setminus A$.
Then $G(A)$ has the [[Beta distribution|0EW824Z7G2]] with parameters $\alpha H(A)$ and $\alpha (1 - H(A))$.
Hence $E[G(A)] = H(A)$ and $\mathrm{Var}(G(A)) = H(A)(1 - H(A))/(\alpha + 1)$.

Non-example: let $H$ be uniform on $[0, 1]$ and let $G = H$ be constant.
Then $G$ is not a Dirichlet process for any $\alpha$.
For $A = [0, 1/2]$ the variance of $G(A)$ is $0$, while a Dirichlet process gives the variance $1/(4(\alpha + 1)) > 0$.

A Dirichlet process exists for every $\alpha$ and $H$ as above.
One construction is stick-breaking.
Let $V_1, V_2, \ldots$ be independent with the Beta distribution with parameters $1$ and $\alpha$, and let $\theta_1, \theta_2, \ldots$ be independent with law $H$ and independent of the $V_i$.
Put $w_k = V_k \prod_{i < k} (1 - V_i)$.
Then $G = \sum_{k \ge 1} w_k \delta_{\theta_k}$ is a Dirichlet process with parameters $\alpha$ and $H$, where $\delta_\theta$ is the [[Dirac measure|099SXAG014]] at $\theta$.
In particular a Dirichlet process is a discrete measure almost surely.

Conjugacy: suppose $G \sim \mathrm{DP}(\alpha, H)$ and, given $G$, the points $x_1, \ldots, x_n$ are independent with law $G$.
Then, given $x_1, \ldots, x_n$, the measure $G$ is a Dirichlet process with concentration $\alpha + n$ and base measure $(\alpha H + \sum_{i=1}^n \delta_{x_i})/(\alpha + n)$.