  We review the class of species sampling models (SSM). In particular, we
investigate the relation between the exchangeable partition probability
function (EPPF) and the predictive probability function (PPF). It is
straightforward to define a PPF from an EPPF, but the converse is not
necessarily true. In this paper we introduce the notion of putative PPFs and
show novel conditions for a putative PPF to define an EPPF. We show that all
possible PPFs in a certain class have to define (unnormalized) probabilities
for cluster membership that are linear in cluster size. We give a new necessary
and sufficient condition for arbitrary putative PPFs to define an EPPF.
Finally, we show posterior inference for a large class of SSMs with a PPF that
is not linear in cluster size and discuss a numerical method to derive its PPF.
