Hierarchical Species Sampling Models

Bassetti, Federico; Casarin, Roberto; Rossini, Luca

doi:10.1214/19-BA1168

This paper introduces a general class of hierarchical nonparametricprior distributions which includes new hierarchical mixture priors such as thehierarchical Gnedin measures, and other well-known prior distributions such asthe hierarchical Pitman-Yor and the hierarchical normalized random measures.The random probability measures are constructed by a hierarchy of generalizedspecies sampling processes with possibly non-diffuse base measures. The proposedframework provides a probabilistic foundation for hierarchical random measures,and allows for studying their properties under the alternative assumptions of dif-fuse, atomic and mixed base measure. We show that hierarchical species samplingmodels have a Chinese Restaurants Franchise representation and can be used asprior distributions to undertake Bayesian nonparametric inference. We provide ageneral sampling method for posterior approximation which easily accounts fornon-diffuse base measures such as spike-and-slab

This paper introduces a general class of hierarchical nonparametric prior distributions which includes 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. 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 probabilistic foundation for hierarchical random measures, and allows for studying their properties under the alternative assumptions of diffuse, atomic and mixed base measure. 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 general sampling method for posterior approximation which easily accounts for non-diffuse base measures such as spike-and-slab.