What does "CRP is a marginalized version of PYP" mean? I've been reading this phrase and I don't know what it means "CRP is a marginalized version of PYP". What are the parameters/latent-variables we are marginalizing out to drive CRP from PYP?
 A: First observe that the Pitman-Yor (or, alternatively, the Poisson-Dirichlet process) can be constructed via stick-breaking. That is, saying
\begin{align}
\beta'_{k=1,\dots,\infty} &\sim \text{Beta}(1-d, \alpha + kd)\\
\beta_{k=1,\dots,\infty} &= \beta'_k \prod_{h=1}^{k-1}(1 - \beta_h')\\
\theta^*_{k=1,\dots,\infty} &\sim H\\
\theta_{i=1,\dots,n} &\sim \sum_{k=1}^\infty \beta_k \delta_{\theta^*_k}(\cdot)
\end{align}
is equivalent to saying
\begin{align}
\theta_{i=1,\dots,n} &\sim G\\
G &\sim \text{PYP}(\alpha, d, H).
\end{align}
The statement "CRP is a marginalized version of PYP" simply refers to the fact that we can marginalize out the atomic weights $\beta_{k=1,\dots,\infty}$. Specifically, we can demonstrate that
$$
P(\theta_{n+1} \mid \theta_{1,\dots,n}, \alpha, d, H) = \frac{\alpha + dm^*}{\alpha + n}H(\theta_{n+1})
+ \sum_{k=1}^{m^*} \frac{m_k - d}{\alpha + n} \delta_{\theta^*_k}(\theta_{n+1}),
$$
where $\theta^*_{k=1,\dots,m^*}$ are the distinct values already observed, and $m_k$ refers to the number of values of $\theta^*_{k}$ that have been observed. This last form is identical to the generalized Chinese restaurant process (CRP), hence the expression.
