# Dirichlet process Posterior notation

While reading through Yee Whye Teh's tutorial on Dirichlet Process, I came across the Posterior distribution for DP. After observing some draws $$\theta_1,\theta_2, ... ,$$, the posterior is defined as $$G | \theta_1,...,\theta_n \sim DP(\alpha + n, \frac{\alpha}{\alpha + n}H +\frac{n}{\alpha + n}\frac{\sum_{i=1}^n \delta_{\theta_i}}{n})$$ I understand the posterior distribution of the DP is thus generated in the following manner: Draw an infinity of $$\rho's$$: $$\rho_1,\rho_2, ...$$ following a $$GEM(\alpha + n)$$ distribution. And then draw an infinity of $$\theta$$'s from the mixture of base distribution $$H$$ and the empirical distribution.

Can someone explain what $$\delta_{\theta_i}$$ means in this case ?

$$\delta_{\theta_{i}}$$ is the dirac measure on $$\theta_{i}$$:
$$\delta_{\theta_{i}}(A) = 1_{A}(\theta_{i})$$
It is present in the posterior of the DP because, for a measurable partition $$B_1,...,B_k$$ of $$\Theta$$, the marginal distribution becomes:
$$(G(B_1),...,G(B_k)) | \theta_{1},...,\theta_{n} \sim DP(\alpha H + \sum_{i=1}^n \delta_{\theta_i}(B_1), ..., \alpha H + \sum_{i=1}^n \delta_{\theta_i}(B_k)) \sim DP(\alpha H + n_1, ..., \alpha H + n_k)$$
since $$n_j = \sum_{i=1}^n 1_{B_j}(\theta_i) = \sum_{i=1}^n \delta_{\theta_i}(B_j)$$