On a Class of Random Probability Measures with General Predictive Structure

Scandinavian Journal of Statistics

Volumn 38, Issue 2, 2011, Pages 359-376

  Favaro, Stefano ^a   Prünster, Igor ^a   Walker, Stephen G ^b  

^a UNIVERSITY OF TURIN   (Italy)

^b UNIVERSITY OF KENT   (United Kingdom)

Author keywords

Bayesian Non parametrics; Dirichlet process; Exchangeable random partitions; Generalized gamma process; Lauricella hypergeometric function; Species sampling models

Indexed keywords

EID: 79955769481     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2010.00702.x     Document Type: Article

References (46)

1. Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 1152-1174.
2. Bell, E. T. (1927). Partition polynomials. Ann. Math. 29, 38-46.
3. Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrsch. verw. Geb. 57, 39-71.
4. Charalambides, C. A. (2005). Combinatorial methods in discrete distributions. Wiley, Hoboken, NJ.
5. Comtet, L. (1974). Advanced combinatorics. D, Reidel Publishing Company, Boston.
6. Dunson, D. B. (2010). Non-parametric Bayes applications to biostatistics. In Bayesian nonparametrics (eds N. L. Hjort, C. C. Holmes, P. Müller & S. G. Walker). pp.223-268. Cambridge University Press, in press.
7. Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theor. Pop. Biol. 3, 87-112.
8. Exton, H. (1976). Multiple hypergeometric functions and application, Ellis Horwood, Chichester.
9. Favaro, S., Prünster, I. & Walker, S. G. (2009). On a generalized Chu-Vandermonde identity. Technical Report, University of Turin.
10. Ferguson, T. S. (1973). A Bayesian analysis of some non-parametric problems. Ann. Statist. 1, 209-230.
11. Ferguson, T. S. & Klass, M. J. (1972). A representation of independent increments processes without Gaussian components. Ann. Math. Statist. 43, 1634-1643.
12. Gnedin, A. & Pitman, J. (2005). Exchangeable Gibbs partitions and Stirling triangles. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325, 83-102.
13. Good, I. J. & Toulmin, G. H. (1956). The number of new species, and the increase in population coverage, when a sample is increased. Biometrika 43, 45-63.
14. Ho, M.-W., James, L. F. & Lau, J. W. (2007). Gibbs partitions (EPPF's) derived from a stable Subordinator are Fox H and Meijer G transforms, Preprint. MatharXiv arXiv:0708.0619v2.
15. Ishwaran, H. & James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96, 161-173.
16. Ishwaran, H. & James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13, 1211-1235.
17. James, L. F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Manuscript. MatharXiv arXiv:math/0205093v1.
18. James, L. F., Roynette, B. & Yor, M. (2008). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5, 346-415.
19. James, L. F., Lijoi, A. & Prünster, I. (2009). Posterior analysis for normalized random measure with independent increments. Scand. J. Statist. 36, 76-97.
20. Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. B 37, 1-22.
21. Kingman, J. F. C. (1993). Poisson processes. Oxford University Press, Oxford.
22. Lijoi, A. & Prünster, I. (2010). Models beyond the Dirichlet process. In Bayesian nonparametrics (eds N. L. Hjort, C. C. Holmes, P. Müller & S. G. Walker). pp80-130. Cambridge University Press, in press.
23. Lijoi, A. & Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32, 1469-1495.
24. Lijoi, A., Mena, R. H. & Prünster, I. (2005). Bayesian nonparametric analysis for a generalized Dirichlet process prior. Statist. Inf. Stoc. Proc. 8, 283-309.
25. Lijoi, A., Mena, R. H. & Prünster, I. (2007a). Controlling the reinforcement in Bayesian non-parametric mixture models. J. Roy. Statist. Soc. B 69, 715-740.
26. Lijoi, A., Mena, R. H. & Prünster, I. (2007b). Bayesian Nonparametric estimation of the probability of discovering a new species. Biometrika 94, 769-786.
27. Lijoi A., Mena, R.H. & Prünster, I. (2007c). A Bayesian nonparametric method for prediction in EST analysis. BMC Bioinform. 8, 339.
28. Lijoi A., Mena, R. H. & Prünster, I. (2008a). A Bayesian nonparametric approach for comparing clustering structures in EST libraries. J. Comput. Biol. 15, 1315-1327.
29. Lijoi, A., Prünster, I. & Walker, S. G. (2008b). Bayesian nonparametric estimators derived from conditional Gibbs structures. Ann. Appl. Probab. 18, 1519-1547.
30. Lijoi, A., Prünster, I. & Walker, S. G. (2008c). Investigating nonparametric priors with Gibbs structure. Statist. Sinica 18, 1653-1668.
31. Lo, A. I. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates. Ann. Statist. 12, 351-357.
32. Lo, A. I. & Weng, C. S. (1989). On a class of Bayesian nonparametric estimates: II. Hazard rate estimates. Ann. Inst. Statist. Math. 41, 227-245.
33. Müller, P. & Quintana, F. A. (2004). Nonparametric Bayesian data analysis. Statist. Sci. 19, 95-110.
34. Müller, P. & Quintana, F. A. (2010a). More nonparametric Bayesian models for biostatistics. In Bayesian nonparametrics (eds N. L. Hjort, C. C. Holmes, P. Müller & S. G. Walker). pp274-290. Cambridge University Press, in press.
35. Müller, P. & Quintana, F. A. (2010b). Random Partition models with regression on covariates. J. Stat. Plann. Inf., doi:10.1016/j.jspi-2010.03.002.
36. Navarrete, C., Quintana, F. A. & Müller, P. (2008). Some issues on nonparametric Bayesian modeling using species sampling models. Stat. Model. 8, 3-21.
37. Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Rel. Fields 102, 145-158.
38. Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, probability and game theory. Papers in honor of David Blackwell (eds T. S. Ferguson, S. L. Shapley & J. B. MacQueen). Lecture Notes Monograph Series 30, 245-267. IMS, Hayward.
39. Pitman, J. (2003). Poisson-Kingman partitions. In Science and statistics: a Festschrift for Terry speed (ed. D. R. Goldstein). Lecture Notes Monograph Series, 40, 1-34. IMS, Beachwood, OH.
40. Pitman, J. (2006). Combinatorial stochastic processes. Ecole d'Eté de Probabilités de Saint-Flour XXXII. Lecture Notes in Mathematics No. 1875, Springer, New York.
41. Regazzini, E. (1998). An example of the interplay between statistics and special functions. In Tricomi's idea and contemporary applied mathematics Acc. Naz. Lincei e Acc. Scienze Torino, Roma, Atti dei Convegni Lincei, Vol. 147, pp. 303-320.
42. Regazzini, E., Lijoi, A. & Prünster, I. (2003). Distributional results for means of random measures with independent increments. Ann. Statist. 31, 560-585.
43. von Renesse, M. K., Yor, M. & Zambotti, L. (2008). Quasi-invariance properties of a class of subordinators. Stoch. Proc. Appl. 118, 2038-2057.
44. Thorin, O. (1977). On the infinite divisibility of the Pareto distribution, Scand Actuar J. 1, 31-40.
45. Walker, S. G. & Damien, P. (2000). Representation of Lévy processes without Gaussian components. Biometrika 87, 477-483.
46. Zabell, S. L. (1982). W. E. Johnson's ''sufficientness'' postulate. Ann. Statist. 10, 1090-1099.