Difference between hierarchical dirichlet process and nested dirichlet process There have some extensions to Dirichlet process. One is Hierarchical Dirichlet process, and another is Nested Dirichlet Process. What are the differences between these two?
I once read the paper of Bayesian Nonparametric Inference -- Why and How. The author discusses their differences as follows, but I am not clear how to understand this. Any more explanations is highly appreciated.

 A: I cannot post a comment with a Figure, so bear with me...
In the NDP manuscript there is a great figure that let me understand exactly what is going on:

In the HDP, different groups are modelled with different mixing distributions (i.e. $P(G_1 = G_2) = 0$) that however share the same set of atoms.
In the NDP different groups are modelled with a Dirichlet Process, meaning that different groups can have the same mixing distribution (i.e $P(G_1 = G_2) > 0$), as in happens in the figure for $G_1$ and $G_3$.
However, if two measures are different, they are completely different (weights and atoms), in contrast with the HDP for which the atoms are equal.
A: For HDP, $G_j$ are samples from a discrete distribution $G_0=\sum_{k=1}^\infty\pi_k\delta_{\theta_k}$, with i.i.d. atoms $\theta_k$ from base measure $H$.
For NDP, $G_j$ are samples from $Q$, which is a little complicated. $Q$ is a sample from a $DP$, whose base measure is another Dirichlet Process $DP(\beta, H)$. Therefore, by definition, $Q=\sum_{k=1}^\infty\pi_k\delta_{G_k^*}$, where the i.i.d. atoms $G_k^*$ are drawn from the $DP(\beta, H)$. In other words, $Q$ is a distribution, with probability mass over samples from a DP. As $G_j\sim Q$, the samples $G_j$ can only be equal to one of the samples from the inner $DP$, i.e. one of the atoms $G_k^*$.
