Hierarchical Species Sampling Models

TL;DR

This paper develops a comprehensive framework for hierarchical nonparametric priors using generalized species sampling processes, enabling advanced Bayesian inference with flexible clustering properties.

Contribution

It introduces a unified hierarchical species sampling model framework, including new priors like hierarchical Gnedin measures, and provides methods for posterior sampling and analysis.

Findings

Hierarchical species sampling models have a Chinese Restaurants Franchise representation.

The framework encompasses well-known priors like hierarchical Pitman-Yor and normalized random measures.

Numerical illustrations demonstrate the applicability of the proposed 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.