21 Mar, 2016
In the Dirichlet Process, DP($\alpha, G_0$), if $x_i$ represents observation $i$, $K_i$ is the number of unique labels before assigning observation $i$, $c_k$ represents the number of observations assigned to label $k$, and $u_k$ is the value for label $k$, then assign observation $i$ to be
\[\begin{cases} x_i \sim G_0, & \text{w.p.} ~~ \ds\frac{\alpha}{\alpha+i-1} \\ \\ x_i = u_k, & \text{w.p.} ~~ \ds\frac{c_k}{\alpha+i-1}, \text{ for } k=1,...,K_i \\ \end{cases}\]In the Pitman-Yor Process, PYP($\delta,\alpha, G_0$), $\delta \in (0,1)$, $\alpha \gt -\delta$,
\[\begin{cases} x_i \sim G_0, & \text{w.p.} ~~ \ds\frac{\alpha+\delta K_i}{\alpha+i-1} \\ \\ x_i = u_k, & \text{w.p.} ~~ \ds\frac{c_k-\delta}{\alpha+i-1}, \text{ for } k=1,...,K_i \\ \end{cases}\]DP($\alpha,G_0$) = PYP($0,\alpha,G_0$). NSP($\delta,G_0$) = PYP($\delta,0,G_0$). NSP = Normalized Stationary Process.
So, in the normalized stationary process, NSP($\delta,G_0$), $\delta \in (0,1)$,
\[\begin{cases} x_i \sim G_0, & \text{w.p.} ~~ \ds\frac{\delta K_i}{i-1} \\ \\ x_i = u_k, & \text{w.p.} ~~ \ds\frac{c_k-\delta}{i-1}, \text{ for } k=1,...,K_i \\ \end{cases}\]Note that in the NSP, the first observation is drawn from the centering distribution $G_0$.