Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as “sharing of information”. In this paper we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman–Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.

Distribution theory for hierarchical processes

Camerlenghi, Federico;Lijoi, Antonio;Pruenster, Igor
2019

Abstract

Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as “sharing of information”. In this paper we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman–Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.
2019
2018
Camerlenghi, Federico; Lijoi, Antonio; Orbanz, Peter; Pruenster, Igor
File in questo prodotto:
File Dimensione Formato  
HierCRM_rev_final.pdf

non disponibili

Descrizione: Articolo principale
Tipologia: Documento in Pre-print (Pre-print document)
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 965.01 kB
Formato Adobe PDF
965.01 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11565/4001763
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 46
  • ???jsp.display-item.citation.isi??? 40
social impact