Are Gibbs-type priors the most natural generalization of the Dirichlet process?

Pierpaolo De Blasi (), Stefano Favaro (), Antonio Lijoi (), Ramsés H. Mena (), Igor Prünster () and Matteo Ruggiero ()
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Pierpaolo De Blasi: Università degli Studi di Torino and Collegio Carlo Alberto
Stefano Favaro: Università degli Studi di Torino and Collegio Carlo Alberto
Antonio Lijoi: Università di Pavia and Collegio Carlo Alberto
Ramsés H. Mena: Universidad Autónoma de México, México
Igor Prünster: Università degli Studi di Torino and Collegio Carlo Alberto
Matteo Ruggiero: Università degli Studi di Torino and Collegio Carlo Alberto

No 54, DEM Working Papers Series from University of Pavia, Department of Economics and Management

Abstract: Discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in Bayesian inference. Indeed, many popular nonparametric priors, such as the Dirichlet and the Pitman–Yor process priors, select discrete probability distributions almost surely and, therefore, automatically induce exchangeable random partitions. Here we focus on the family of Gibbs–type priors, a recent and elegant generalization of the Dirichlet and the Pitman–Yor process priors. These random probability measures share properties that are appealing both from a theoretical and an applied point of view: (i) they admit an intuitive characterization in terms of their predictive structure justifying their use in terms of a precise assumption on the learning mechanism; (ii) they stand out in terms of mathematical tractability; (iii) they include several interesting special cases besides the Dirichlet and the Pitman–Yor processes. The goal of our paper is to provide a systematic and unified treatment of Gibbs–type priors and highlight their implications for Bayesian nonparametric inference. We will deal with their distributional properties, the resulting estimators, frequentist asymptotic validation and the construction of time–dependent versions. Applications, mainly concerning hierarchical mixture models and species sampling, will serve to convey the main ideas. The intuition inherent to this class of priors and the neat results that can be deduced for it lead one to wonder whether it actually represents the most natural generalization of the Dirichlet process.

Keywords: Bayesian Nonparametrics; Clustering; Consistency; Dependent process; Discrete nonparametric prior; Exchangeable partition probability function; Gibbs–type prior; Pitman–Yor process; Mixture model; Population Genetics; Predictive distribution; Species sampling. (search for similar items in EconPapers)
Pages: 45 pages
Date: 2013-10
New Economics Papers: this item is included in nep-dcm and nep-ecm
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