Summary
Call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn 1,...,n k, the probability that the partition breaks the firstn 1+...+nk integers intok particular classes, of sizesn 1,...,nk in order of their first elements, has the same valuep(n 1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n 1,...nk). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure.
This is a preview of subscription content, log in to check access.
References
- 1.
Aldous, D.J.: Exchangeability and related topics. In: Hennequin, P.L. (ed.) École d'Été de Probabilités de Saint-Flour XII. (Lect. Notes Math. vol. 1117) Berlin Heidelberg New York: Springer 1985
- 2.
Antoniak, C.: Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Stat.2, 1152–1174 (1974)
- 3.
Blackwell, D., MacQueen, J.B.: Ferguson distributions via Pólya urn schemes. Ann. Stat.1, 353–355 (1973)
- 4.
de Finetti, B.: Sur la condition d'équivalence partielle. Actualités Scientifiques et Industrielles. 739 (1938). Herman and Cie: Paris. Translated In: Studies in Inductive and Probability, II. Jeffrey, R. (ed.) University of California Press: Berkeley 1980
- 5.
Diaconis, P., Freedman, D.: Partial exchangeability and sufficiency. In: Ghosh, J.K., Roy, J. (eds.) Statistics Applications and New Directions; Proceedings of the Indian Statistical Institute Golden Jubilee International Conference; Sankhya A. pp. 205–236 Indian Statistical Institute 1984
- 6.
Donnelly, P.: Partition structures, Pólya urns, the Ewens sampling formula, and the ages of alleles. Theoret. Population Biology30, 271–288 (1986)
- 7.
Donnelly, P.: The heaps process, libraries and size biased permutations. J. Appl. Probab.28, 322–335 (1991)
- 8.
Donnelly, P., Joyce, P.: Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stochast. Processes Appl.31, 89–103 (1989)
- 9.
Donnelly, P., Tavaré, S.: The ages of alleles and a coalescent. Adv. Appl. Probab.18, 1–19, 1023 (1986)
- 10.
Engen, S.: Stochastic abundance models with emphasis on biological communities and species diversity. London: Chapman and Hall Ltd., 1978
- 11.
Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol.3, 87–112 (1972)
- 12.
Ewens, W.J.: Population genetics theory—the past and the future. In: Lessard, S. (ed.), Mathematical and statistical problems in evolution. Montreal: University of Montreal Press 1988
- 13.
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)
- 14.
Hoppe, F.M.: Pólya-like urns and the Ewens sampling formula. J. Math. Biol.20, 91–94 (1984)
- 15.
Hoppe, F.M.: Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab.23, 1008–1012 (1986)
- 16.
Hoppe, F.M.: The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol.25, 123–159 (1987)
- 17.
Ignatov, T.: On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures. Theory Probab. Appl.27, 136–147 (1982)
- 18.
Johnson, W.E.: Probability: the deductive and inductive problems. Mind49, 409–423 (1932)
- 19.
Kingman, J.F.: The population structure associated with the Ewens sampling formula. Theor. Popul. Biol.11, 274–283 (1977)
- 20.
Kingman, J.F.: Random partitions in population genetics. Proc. R. Soc. Lond. A.361, 1–20 (1978)
- 21.
Kingman, J.F.: The representation of partition structures. J. London Math. Soc.18, 374–380 (1978)
- 22.
Kingman, J.F.: The coalescent. Stochast. Processes Appl.13, 235–248 (1982)
- 23.
McCloskey, J.W.: A model for the distribution of individuals by species in an environment. Ph. D. Thesis, Michigan State University (1965)
- 24.
Patil, G.P., Taillie, C.: Diversity as a concept and its implications for random communities. Bull. Int. Stat. Inst.XLVII, 497–515 (1977)
- 25.
Perman, M., Pitman, J., Yor, M.: Size-biased sampling of Poisson point processes and excursions. Probab. Related Fields92, 21–39 (1992)
- 26.
Pitman, J.: Partition structures derived from Brownian motion and stable subordinators. Technical Report 346, Dept. Statistics, U.C. Berkeley Preprint (1992)
- 27.
Pitman, J.: Random discrete distributions invariant under size-biased permutation. Technical Report 344, Dept. Statistics, U.C. Berkeley (1992) To appear in J. Appl. Probab.
- 28.
Pitman, J.: The two-parameter generalization of Ewens' random partition structure. Technical Report 345, Dept. Statistics, U.C. Berkeley, (1992)
- 29.
Pitman, J., Yor, M.: The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Preprint (1994)
- 30.
Watterson, G.A.: The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Probab.13, 639–651 (1976)
- 31.
Zabell, S.L.: Predicting the unpredictable. Synthese90, 205–232 (1992)
- 32.
Zabell, S.L.: The continuum of inductive methods revisited. Preprint (1994)
Author information
Affiliations
Additional information
Research supported by N.S.F. Grants MCS91-07531 and DMS-9404345
Rights and permissions
About this article
Cite this article
Pitman, J. Exchangeable and partially exchangeable random partitions. Probab. Th. Rel. Fields 102, 145–158 (1995). https://doi.org/10.1007/BF01213386
Received:
Accepted:
Issue Date:
Mathematics Subject Classification
- 60G09
- 60C05
- 60J50