Hierarchical Species Sampling Models

Abstract

This paper introduces a general class of hierarchical nonparametric prior distributions. The random probability measures are constructed by a hierarchy of generalized species sampling processes with possibly non-diffuse base measures. The proposed framework provides a general probabilistic foundation for hierarchical random measures with either atomic or mixed base measures and allows for studying their properties, such as the distribution of the marginal and total number of clusters. We show that hierarchical species sampling models have a Chinese Restaurants Franchise representation and can be used as prior distributions to undertake Bayesian nonparametric inference. We provide a method to sample from the posterior distribution together with some numerical illustrations. Our class of priors includes some new hierarchical mixture priors such as the hierarchical Gnedin measures, and other well-known prior distributions such as the hierarchical Pitman-Yor and the hierarchical normalized random measures.

Hierarchical Normalized Completely Random Measures to Cluster Grouped Data

“…However, the eppf's in formula (11) involve the computation of integrals that depend on the two Lévy intensities α and α 0 , see (5). In order to avoid the computation of such integrals, we resort to a standard approach for NormCRMs (see 20).…”

“…In Bayesian nonparametrics, such hierarchical structure has been used to introduce the celebrated hierarchical Dirichlet process (30; 31), with successful applications in genetics, image segmentation and topic modeling, to mention a few (6; teh). More recently, hierarchical processes have been investigated from an analytical perspective by (8; 7), while (5) have focused on hierarchical species sampling models. These authors have shown that extensions to normalized completely random measures encompassing the Dirichlet process allow for richer predictive structures.…”

“…With respect to the recent contributions on Bayesian nonparametric hierarchical processes of (7) and (5), which investigate more theoretical aspects, in this paper we provide a detailed study of the two-layered hierarchical clustering structure induced by these models.…”

Perfect Sampling of the Posterior in the Hierarchical Pitman–Yor Process

“…We refer to the recent works of and Bassetti et al (2020) for a detailed study of the predictive probability (1.2) and related results. Teh et al (2006) proposed a strategy for sampling from the posterior distribution of the hierarchical Dirichlet process and from the posterior distribution of the hierarchical Pitman-Yor process.…”

“…See Teh and Jordan (2010) for a review on hierarchical nonparametric priors. In Bayesian nonparametrics, theoretical developments and applications of the hierarchical Pitman-Yor process have been considered in language modeling (Teh, 2006;Huang and Renals, 2007;Wood et al, 2009), infinite hidden Markov modeling (Beal et al, 2002;Van Gael et al, 2008;Blunsom and Cohn, 2011), species sampling with multiple populations (Battiston et al, 2018;Bassetti et al, 2020;, clustering (Argiento et al, 2020), graphical modeling (Creamschi et al, 2020), image segmentation (Sudderth and Jordan, 2009), and topic models (Sato and Nakagawa, 2010;Araki et al, 2012;Lindsey et al, 2012). In this paper we evaluate and compare multiple computational strategies for posterior inference under the hierarchical Pitman-Yor process prior.…”