On some distributional properties of hierarchical processes

Vectors of hierarchical random probability measures are popular tools in Bayesian nonparametrics. They may be used as priors whenever partial exchangeability is assumed at the level of either the observations or of some latent variables involved in the model. The first contribution in this direction can be found in Teh et al. (2006), who introduced the hierarchical Dirichlet process. Recently, Camerlenghi et al. (2018) have developed a general distribution theory for hierarchical processes, which includes the derivation of the partition structure, the posterior distribution and the prediction rules. The present paper is a review of these theoretical findings for vectors of hierarchies of Pitman--Yor processes.