Abstract
Conditional Gibbs partitions naturally arise in Bayesian nonparametric analysis of species sampling problems under almost surely discrete priors inducing infinite exchangeable partitions with distribution in Gibbs product form. In this setting interest relies on posterior predictive inference on some characteristics of a population of species, given an initial sample of observations. Here we focus on the subclass of Poisson-Kingman partitions driven by the Stable subordinator, and, relying on the unconditional theory of exchangeable Gibbs partitions, derive some additional results for the posterior partition, the conditional α diversity and a Stirling’s approximation of the Gibbs weights.
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References
Cerquetti, A.: A note on Bayesian nonparametric priors derived from exponentially tilted Poisson-Kingman models. Stat. Probab. Lett. 77(18), 1705–1711 (2007)
Cerquetti, A.: On a Gibbs characterization of normalized generalized Gamma processes. Stat. Probab. Lett. 78, 3123–3128 (2008)
Cerquetti, A.: A generalized sequential construction of exchangeable Gibbs partitions with application. In: Proceedings of S.Co. 2009. Milano, Italy (2009)
Cerquetti, A.: A decomposition approach to Bayesian nonparametric estimation under two-parameter Poisson-Dirichlet priors. In: Proceedings of ASMDA 2011, Rome, Italy (2011)
Cerquetti, A.: Stirling’s approximations for exchangeable Gibbs weights. arXiv:1206.6812v1 [math.PR] (2012)
Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87–112 (1972)
Favaro, S., Lijoi, A., Mena, R.H., Prünster, I.: Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior. J. Roy. Stat. Soc. B 71, 993–1008 (2009)
Favaro, S., Lijoi, A., Prünster, I.: Asymptotics for a Bayesian nonparametric estimator of species variety. 18, 1267–1283 Bernoulli (2012a)
Favaro, S., Lijoi, A., Prünster, I.: On the stick breaking representation of normalized inverse Gaussian priors. 9, 663–674 Biometrika (2012b)
Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)
Fisher, R.A., Corbet, A.S., Williams, C.B.: The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 12, 42–58 (1943)
Gnedin, A., Haulk, S., Pitman, J.: Characterizations of exchangeable partitions and random discrete distributions by deletion properties. arXiv:0909.3642 [math:PR] (2009)
Gnedin, A., Pitman, J.: Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. 138(3), 5674–5685 (2006)
Ho, M.-W., James, L.F., Lau, J.W.: Gibbs partitions (EPPF’s) derived from a stable subordinator are Fox H – And Meijer G – Transforms. arXiv:0708.0619v2 [math.PR] (2007)
Hsu, L.C., Shiue, P.J.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20, 366–384 (1998)
James, L.F., Lijoi, A., Prünster, I.: Posterior analysis for normalized random measures with independent increments. Scand. J. Stat. 36, 76–97 (2009)
Kingman, J.F.C.: The representation of partition structures. J. Lond. Math. Soc. 18, 374–380 (1978)
Lijoi, A., Mena, R.H., Prünster, I.: Hierarchical mixture modeling with normalized Inverse Gaussian priors. J. Am. Stat. Assoc. 100, 1278–1291 (2005)
Lijoi, A., Mena, R.H., Prünster, I.: Controlling the reinforcement in Bayesian non-parametric mixture models. J. Roy. Stat. Soc. B 69(4), 715–740 (2007a)
Lijoi, A., Mena, R.H., Prünster, I.: Bayesian nonparametric estimation of the probability of discovering new species. Biometrika 94, 769–786 (2007b)
Lijoi, A., Prünster, I., Walker, S.G.: Bayesian nonparametric estimators derived from conditional Gibbs structures. Ann. Appl. Probab. 18, 1519–1547 (2008a)
Lijoi, A., Prünster, I., Walker, S.G.: Investigating nonparametric priors with Gibbs structure. Statistica Sinica 18, 1653–1668 (2008b)
Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theor. Rel. Fields 102, 145–158 (1995)
Pitman, J.: Some developments of the Blackwell-MacQueen urn scheme. In: Ferguson, T.S., Shapley, L.S., MacQueen, J.B. (eds.) Statistics, Probability and Game Theory. IMS Lecture Notes-Monograph Series, vol. 30, pp. 245–267. Institute of Mathematical Statistics, Hayward (1996)
Pitman, J.: Brownian motion, bridge, excursion, and meander characterized by sampling at uniform times. Elect. J. Prob. 4, 1–33 (1999)
Pitman, J.: Poisson-Kingman partitions. In: Goldstein, D.R. (ed.) Science and Statistics: A Festschrift for Terry Speed. Lecture Notes-Monograph Series, vol. 40, pp. 1–34. Institute of Mathematical Statistics, Hayward (2003)
Pitman, J.: Combinatorial stochastic processes. Ecole d’Eté de Probabilité de Saint-Flour XXXII – 2002. Lecture Notes in Mathematics N. 1875. Springer, Berlin (2006)
Pitman, J., Yor, M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997)
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The author is partially supported by grant PRIN MIUR:2008CEFF37.
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Cerquetti, A. (2013). Some Contributions to the Theory of Conditional Gibbs Partitions. In: Grigoletto, M., Lisi, F., Petrone, S. (eds) Complex Models and Computational Methods in Statistics. Contributions to Statistics. Springer, Milano. https://doi.org/10.1007/978-88-470-2871-5_7
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