title: Defining Predictive Probability Functions for Species Sampling Models.
abstract: 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.
authors.affiliation: Department of Statistics, Seoul National University, Seoul, 151-747, South Korea ( leejyc@gmail.com ).
authors.forename: Jaeyong
authors.initials: J
authors.lastname: Lee
authors.affiliation: Departamento de Estadística, Pontificia Universidad Católica de Chile, Macul, Santiago 22, Chile ( quintana@mat.puc.cl ).
authors.forename: Fernando A
authors.initials: FA
authors.lastname: Quintana
authors.affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1202, USA ( pmueller@math.utexas.edu ).
authors.forename: Peter
authors.initials: P
authors.lastname: Müller
authors.affiliation: Department of Biostatistics, Harvard University, and Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute, Boston, Massachusetts 02115, USA ( ltrippa@jimmy.harvard.edu ).
authors.forename: Lorenzo
authors.initials: L
authors.lastname: Trippa
keywords: Species sampling prior; exchangeable partition probability functions; prediction probability functions