Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 12 (2007) > Paper 40 open journal systems 


Record indices and age-ordered frequencies in Exchangeable Gibbs Partitions

Robert C. Griffiths, University of Oxford
Dario Spanò, University of Oxford


Abstract
The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence of record indices (i.e. the least elements) of the blocks of the partition. In particular we show that, conditionally on the record indices, the distribution of the age-ordered frequencies has a left-neutral stick-breaking structure. Such a property in fact characterizes the Gibbs family among all exchangeable partitions and leads to further interesting results on: (i) the conditional Mellin transform of the k-th oldest frequency given the k-th record index, and (ii) the conditional distribution of the first k normalized frequencies, given their sum and the k-th record index; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens' partitions.


Full text: PDF

Pages: 1101-1130

Published on: August 25, 2007
Bibliography
  1. D.J. Aldous. Exchangeability and related topics, volume 1117 of Lecture Notes in Mathematics, pp. 1--198. Springer-Verlag, Berlin, 1985. Lecture notes from Ecole d'été de Probabilités de Saint-Flour XIII - 1983.MR 88d:60107
  2. J. Bertoin. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, vol. 102. Cambridge University Press, Cambridge 2006. Math. Review in process.
  3. N. Berestycki and J.Pitman. Gibbs distributions for random partitions generated by a fragmentation process, 2005. http://arXiv.org/abs/math/0512378
  4. K.Bobecka and J. Wesol owski. The Dirichlet distribution and process through neutralities, J. Theor. Probab.,20(2007),295--308. Math. review not available.
  5. R.J. Connor and J.E. Mosimann. Concepts of independence for proportions with a generalization of the Dirichlet distribution, J. Am. Stat. Assoc., 64(1969),194--206.MR 39 #2240
  6. K. Doksum. Tailfree and neutral random probabilities and their posterior distributions, Ann. Probab., 2 (1974), 183--201. MR 51 #9283
  7. R. Dong, C. Goldschmidt, and J. B. Martin. Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees, Ann. Appl. Probab., 16 (2006), 1733--1750. MR2288702
  8. P. Donnelly. Partition structures, Pòlya urns, the Ewens sampling formula, and the ages of alleles Theoret. Population Biol., 30(2) (1986), 271--288. MR0865115
  9. P. Donnelly and P. Joyce. Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex, Stoc. Proc. Appl., 31 (1)(1989), 89--103. MR0996613
  10. P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models, Ann. Probab. 27(1)(1999), 166--205. MR1404525
  11. A. V. Gnedin. The representation of composition structures Ann. Probab., 25(3)(1997), 1437--1450. MR1457625
  12. A. V. Gnedin. On convergence and extensions of size-biased permutations,, J. Appl. Probab., 35(3)(1998), 642--650. MR1659532
  13. A. V. Gnedin. Constrained exchangeable partitions, 2006. To appear in Discr. Math. Comp. Sci.. http://arXiv:math/0608621v1
  14. A. V. Gnedin and J. Pitman. Exchangeable Gibbs partitions and Stirling triangles, Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 325(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody.), 12 (2005), 83--102, 244--245. MR2160320
  15. R. C. Griffiths and S. Lessard. Ewens' sampling formula and related formulae: Combinatorial proofs, extensions to variable population size and applications to ages of alleles, Theor. Popul. Biol., 68(2005), 167--177. mathematical review not available
  16. F. M. Hoppe. Pòlya-like urns and the Ewens' sampling formula, J. Math. Biol., 20(1)(1984), 91--94. MR758915
  17. H. Ishwaran and L.F. James. Some further developments for stick-breaking priors: finite and infinite clustering and classification , Sankhya. The Indian Journal of Statistics, 65(3)(2003), 577--592. 2060608
  18. S. V. Kerov.. Combinatorial examples in the theory of AF-algebras , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 172(Differentsialnaya Geom. Gruppy Li i Mekh.), 10(3)(1989), 55--67, 169--170. 1015698
  19. S. V. Kerov. Subordinators and permutation actions with quasi-invariant measure , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 223(Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I)(1995),81--218, 340. 1374320
  20. S. V. Kerov and N.V. Tsilevich. A random subdivision of an interval generates virtual permutations with the Ewens distribution , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 223(Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I)(1995), 162--180, 339--340. 1374319
  21. J. F. C. Kingman. On the genealogy of large populations,, J. Appl. Probab., 19A(1982), 27--43. MR633178
  22. A. Lijoi, I. Pruenster and S. G. Walker. Bayesian nonparametric estimators derived from conditional Gibbs structures,, Tech. Report, Università degli Studi di Torino, 2007. Mathematical review not available.
  23. S. Nacu. Increments of random partitions, 2004 http://arXiv:math/0310091v2
  24. J. Pitman. Exchangeable and partially exchangeable random partitions, Probab. Theory Related Fields, 102(2)(1995), 145--158. MR1337249
  25. J. Pitman. Random discrete distributions invariant under size-biased permutation, Adv. in Appl. Probab., 28(2)(1996), 525--539. MR1387889
  26. J. Pitman. Some developments of the Blackwell-MacQueen urn scheme, Statistics, probability and game theory, volume 30 of IMS Lecture Notes Monogr. Ser., Inst. Math. Statist., Hayward, CA, 245--267, 1996, MR1481784
  27. J. Pitman. Poisson-Kingman partitions, Statistics and science: a Festschrift for Terry Speed, volume 40 ofIMS Lecture Notes Monogr. Ser., Inst. Math. Statist., Beachwood, OH, pp. 1--34, 2003, MR2004330
  28. J. Pitman. Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 2006. Lecture notes from Ecole d'été de Probabilités de Saint-Flour XXXII - 2002.MR 2245368
  29. S. Walker and P. Muliere. A characterization of a neutral to the right prior via an extension of Johnson's sufficientness postulate, Ann. Statist., 27(2)(1999), 589--599. MR1714716
  30. G. Watterson. Lines of descent and the coalescent, Theor. Popul. Biol., 26(1)(1984), 77--92. MR760232
  31. S. Zabell. Predicting the unpredictable, Synthese, 90(2)(1992), 205--232. MR1148566
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489