Dirichlet process

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 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 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 $(\alpha H + \sum_{i=1}^n \delta_{x_i})/(\alpha + n)$.