A Generalized Sequential Construction of Exchangeable Gibbs Partitions with Application.
Annalisa Cerquetti · 2009
In two recent papers (Lijoi et al. 2007, 2008) a Bayesian prior to posterior analysis for the subclass of exchangeable partitions in Gibbs form of type $\\alpha$ in (0, 1), first introduced in Pitman (2003), and largely studied in Gnedin and Pitman (2006), has been proposed for a nonparametric treatment of the typical species sampling inferential problem, in which a random sample is drawn from an hypothetical infinite population of individuals of various species, whose total number is unknown, and interest lies in conditional inference about future partitions induced by future samples. Here, by resorting to sequential constructions of exchangeable random partitions, and exploiting some known facts about generalized Stirling numbers, we derive a generalized group sequential construction of exchangeable partitions in Gibbs form of type $\\alpha$ to place this theory in its natural probabilistic framework and to provide new insights on the derivation of relevant results. Our construction, which relies on known results of Pitman (2003, 2006) and Gnedin and Pitman (2006), has potential applications for investigating additional distributional results for quantities of statistical interest when exploiting in a Bayesian nonparametric perspective the theory of exchangeable partitions. References Cerquetti, A. (2008) Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results. arXiv:0805.3853v1 [math.PR] Gnedin, A. and Pitman, J. (2006) Exchangeable Gibbs partitions and Stirling triangles. Journal of Mathematical Sciences, 138, 3, 5674–5685. Hsu, L.C. & Shiue, P.J. (1998) A unified approach to generalized Stirling numbers. Adv. Appl. Math., 20, 366-384. Lijoi, A., Mena, R. and Pruenster, I. (2007) Bayesian nonparametric estimation of the probability of discovering new species. Biometrika, 94, 769–786. Lijoi, A., Pruenster, I. and Walker, S.G. (2008) Bayesian nonparametric estimator derived from conditional Gibbs structures. Annals of Applied Probability, 18, 1519-1547. Pitman, J. (2003) Poisson-Kingman partitions. In D.R. Goldstein, editor, Science and Statistics: A Festschrift for Terry Speed, volume 40 of Lecture Notes-Monograph Series, pages 1–34. Institute of Mathematical Statistics, Hayward, California. Pitman, J. (2006) Combinatorial Stochastic Processes. Ecole d’Ete de Probabilite de Saint-Flour XXXII - 2002. Lecture Notes in Mathematics N. 1875, Springer.