A Unified Approach to Hierarchical Random Measures

Abstract

Abstract Hierarchical models enjoy great popularity due to their ability to handle heterogeneous groups of observations by leveraging on their underlying common structure. In a Bayesian nonparametric framework, the hierarchy is introduced at the level of group-specific random measures, and then translated to the observations’ level via suitable transformations. In this work, we propose a new strategy to derive closed-form expressions for the marginal and posterior distributions of each group. Indeed, by directly inserting a suitable set of latent variables into the generative model for the data, we unravel a common core shared by the different hierarchical constructions proposed in the Bayesian nonparametric literature. Specifically, we identify a key identity that underlies these models and highlight its role in the derivation of quantities of interest.