  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.
