Abstract: We congratulate Federico Camerlenghi, David Dunson, Antonio Lijoi, Igor Pruenster and Abel Rodrıguez, from now on referred to as CDLPR, for an an interesting paper. CDLPR are indeed to be commended by a very fine work. By focusing on partitions they uncovered a critical degeneracy issue underlying the nested Dirichlet process (NDP) and other discrete nested processes. The key tool to understand the problem is the partially exchangeable partition probability function (pEPPF), that describes the probability model on partitions induced by models such as the NDP. CDLPR explicitly find the pEPPF in the case of d = 2 samples, showing that it is expressed as a mixture of patterns that may be described as partially exchangeable (both samples are marginally exchangeable, arising from different discrete random probability measures) and fully exchangeable (both samples arise from a common discrete random measure). Specifi- cally, the pEPPF is a convex combination of these two forms. However, if it so happens that one atom is shared by these two random measures, then the structure degenerates to the fully exchangeable case. This is certainly a limitation of nested processes. As discussed in the manuscript, the problem is not restricted to the NDP, but will mani- fest itself in the case of any nested discrete random measure. CDLPR illustrate further this point with real data examples and synthetic data simulations. Solving the degen- eracy problem motivated the (very clever) introduction of their latent nested process (LNP) approach. By allowing idiosyncratic and shared components, CDLPR overcome the degeneracy while retaining modeling flexibility. Essentially, the shared component can explicitly provide atoms in the mixture that are common to both samples, while the idiosyncratic components can adjust to local behavior without being forced to combine all of the mass in a single random measure. Their specific construction involves three random measures with which they create the LNP by normalizing the sum of idiosyn- cratic (μl) and shared measures (μS) thus yielding p ̃l, as given in (14), where l = 1,2. The structure is emphasized by noting that each p ̃l can be expressed as a convex com- bination of normalized versions of μl and μS. Indispensable to the proposal is the fact that flexibility does not come at the expense of practical tractability, which is of course vital for the practical success of models based on LNPs.
We discuss next some possible extensions to the model constructed by CDLPR. (Paper)