Abstract
After brief introduction of the multiplicative measure, defined as a family of measures on integer partitions, which include typical combinatorial structures, this chapter introduces the exponential structure, which plays important roles in statistical inference. It then introduces the Gibbs partition, a generalization of the exponential structure. The generalization is achieved by systematic use of partial Bell polynomials. Gibbs partitions characterize prior processes in Bayesian nonparametrics, and appear as statistical models of diversity in count data. The Ewens sampling formula and the Pitman partition are well-known examples of Gibbs partitions. Finally, this section discusses the asymptotic behaviors of extremes of the sizes of parts in Gibbs partitions. Some of the results are derived by simple analytic approaches.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Logan, B.F., Shepp, L.A.: A variational problem for random Young tableaux. Adv. Math. 26, 206–222 (1977)
Vershik, A.M., Kerov, S.V.: Asymptotic behavior of the Plancherel measure of the symmetric group and the limit from of Young tableaux. Dokl. Akad. Nauk SSSR 233, 1024–1037 (1977)
Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, vol. 219. American Mathematical Society, Providence (2003). Translations of Mathematical Monograph
Borodin, A., Corwin, I.: Macdonald processes. Probab. Theor. Relat. Fields 158, 225–400 (2014)
Vershik, A.M.: Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl. 30, 90–105 (1996)
Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, vol. 2. Cambridge University Press, New York (1987)
Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics. Progress in Mathematics, vol. 244, pp. 525–596. Birkhäuser, Boston (2006)
Takemura, A., et al.: Special issue: statistical theory of statistical disclosure control problem. Proc. Inst. Stat. Math. 51, 181–388 (2003)
Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich (2003). EMS Monographs in Math
Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York (1999)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Flajolet, P., Soria, M.: Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theor. Ser. A 53, 165–182 (1990)
Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.: On the Lambert W function. In: Advances in Computational Mathematics, vol. 5, pp. 329–359. Springer, Heidelberg (1996)
Katz, L.: Probability of indecomposability of a random mapping function. Ann. Math. Stat. 26, 512–517 (1955)
Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87–112 (1972)
Antoniak, C.: Mixture of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 1152–1174 (1974)
Sibuya, M.: A random-clustering process. Ann. Inst. Stat. Math. 45, 459–465 (1993)
Tavaré, S., Ewens, W.J.: The Ewens sampling formula. In: Johnson, N.L., Kotz, S., Balakrishnan, N. (eds.) Multivariate Discrete Distributions. Wiley, New York (1997)
Crane, H.: The ubiquitous Ewens sampling formula. Stat. Sci. 31, 1–19 (2016)
Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)
Pitman, J.: Combinatorial stochastic processes. Ecole d’Été de Probabilités de Saint Flour, Lecture Notes in Mathematics, vol. 1875. Springer, Heidelberg (2006)
Comtet, L.: Advanced Combinatorics. Ridel, Dordrecht (1974)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)
Okano, T., Okuto, Y., Shimizu, A., Niikura, Y., Hashimoto, Y., Yamada, H. The formula of Faà di Bruno and its applications I. Annual Review 2000, Institute of National Sciences, Nagoya City University, pp. 35–44 (2000). in Japanese
Kolchin, V.F.: A problem on the distribution of particles among cells, and cycles of random permutations. Teor. Veroyatnost. i Primenen. 16, 67–82 (1971)
Kerov, S.V: Coherent random allocations, and the Ewens-Pitman formula. Zap. Nauchn. Semi. POMI 325, 127–145 (1995); English translation: J. Math. Sci. 138, 5699–5710 (2006)
Hoshino, N.: Random partitioning over a sparse contingency table. Ann. Inst. Stat. Math. 64, 457–474 (2012)
Fisher, R.A., Corbet, A.S., Williams, C.B.: The relationship 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)
Hoshino, N., Takemura, A.: Relationship between logarithmic series model and other superpopulation models useful for microdata discrosure risk assesment. J. Jpn. Stat. Soc. 28, 125–134 (1998)
Bethlehem, J.G., Keller, W.J., Pannekoek, J.: Disclosure control of microdata. J. Am. Stat. Assoc. 85, 38–45 (1990)
Hoshino, N.: On a limiting quasi-multinomial distribution. Discussion Paper CIRJE-F-361, Center for International Research on the Japanese Economy, Faculty of Economics, The University of Tokyo (2005)
Diaconis, P., Lam, A.: A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40, 1861–1896 (2012)
Charalambides, C.A.: Combinatorial Methods in Discrete Distributions. Wiley, New Jersey (2005)
Pitman, J.: Exchangeable and partially exchangeable random partitions. Probab. Theor. Relat. Fields 102, 145–158 (1995)
Hoshino, N.: A discrete multivariate distribution resulting from the law of small numbers. J. Appl. Probab. 43, 852–866 (2006)
Hoshino, N.: Engen’s extended negative binomial model revisited. Ann. Inst. Stat. Math. 57, 369–387 (2005)
Fisher, R.A.: Tests of significance in harmonic analysis. Proc. Roy. Soc. Lond. Ser. A 125, 54–59 (1929)
Goncharov, V.L.: Some facts from combinatorics. Izv. Akad. Nauk SSSR, Ser. Mat. 8, 3–48 (1944)
Shepp, L.A., Lloyd, S.P.: Ordered cycle length in random permutation. Trans. Am. Math. Soc. 121, 340–357 (1966)
Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press, New York (1955)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New Jersey (1999)
Panario, D., Richmond, B.: Smallest components in decomposable structures: exp-log class. Algorithmica 29, 205–226 (2010)
Arratia, R., Barbour, A.D., Tavaré, S.: Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2, 519–535 (1992)
Feng, S.: The Poisson–Dirichlet Distributions and Related Topics. Springer, Heidelberg (2010)
Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)
Hwang, H.-K.: Asymptotic expansions for Stirling’s number of the first kind. J. Combin. Theor. Ser. A 71, 343–351 (1995)
Mano, S.: Extreme sizes in the Gibbs-type random partitions. Ann. Inst. Stat. Math. 69, 1–37 (2017)
Mano, S.: Partition structure and the \(A\)-hypergeometric distribution associated with the rational normal curve. Electron. J. Stat. 11, 4452–4487 (2017)
Pitman, J.: Brownian motion, bridge, excursion and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4, 33pp (1999)
Keener, R., Rothman, E., Starr, N.: Distribution of partitions. Ann. Stat. 15, 1466–1481 (1978)
Korwar, R.M., Hollander, M.: Contribution to the theory of Dirichlet process. Ann. Probab. 1, 705–711 (1973)
Pollard, H.: The representation of \(e^{-x^\lambda }\) as a Laplace integral. Bull. Am. Math. Soc. 52, 908–910 (1946)
Watterson, G.A.: The sampling theory of selectively neutral alleles. Adv. Appl. Probab. 6, 463–488 (1974)
Yamato, H.: Edgeworth expansions for the number of distinct components associated with the Ewens sampling formula. J. Jpn. Stat. Soc. 43, 17–28 (2013)
Yamato, H.: Poisson approximations for sum of Bernoulli random variables and its application to Ewens sampling formula. J. Jpn. Stat. Soc. 47, 187–195 (2018)
Hansen, J.C.: A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27, 28–43 (1990)
Tsukuda, K.: Functional central limit theorems in \(L^2(0,1)\) for logarithmic combinatorial assemblies. Bernoulli 24, 1033–1052 (2018)
Feng, S.: Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula. Ann. Appl. Probab. 17, 1570–1595 (2007)
Tsukuda, K.: Estimating the large mutation parameter of the Ewens sampling formula. J. Appl. Probab. 54, 42–54 (2017)
Tsukuda, K.: On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size. arXiv: 1704.06768
Griffiths, R.C.: On the distribution of points in a Poisson process. J. Appl. Probab. 25, 336–345 (1988)
Pitman, J., Yor, M.: The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–899 (1997)
Dickman, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat., Astronomi och Fysik. 22, 1–14 (1930)
Handa, K.: The two-parameter Poisson–Dirichlet point process. Bernoulli 15, 1082–1116 (2009)
Griffiths, R.C.: On the distribution of allele frequencies in a diffusion model. Theor. Popul. Biol. 15, 140–158 (1979)
Buchstab, A.A.: An asymptotic estimation of a general number-theoretic function. Mat. Sb. 44, 1239–1246 (1937)
Arratia, R., Tavaré, S.: Limit theorems for combinatorial structures via discrete process approximations. Random Structures Algorithms 3, 321–345 (1992)
Sibuya, M.: Generalized hypergeometric, digamma and trigamma distributions. Ann. Inst. Stat. Math. 31, 373–390 (1979)
Devroye, L.: A triptych of discrete distributions related to the stable law. Stat. Probab. Lett. 18, 349–351 (1993)
Karlin, S.: Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17, 373–401 (1967)
Rouault, A.: Lois de Zipf et sources markoviennes. Ann. Inst. H. Poincaré Sect. B 14, 169–188 (1978)
Yamato, H., Sibuya, M.: Moments of some statistics of Pitman sampling formula. Bull. Inf. Cybernet. 32, 1–10 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 The Author(s)
About this chapter
Cite this chapter
Mano, S. (2018). Measures on Partitions. In: Partitions, Hypergeometric Systems, and Dirichlet Processes in Statistics. SpringerBriefs in Statistics(). Springer, Tokyo. https://doi.org/10.1007/978-4-431-55888-0_2
Download citation
DOI: https://doi.org/10.1007/978-4-431-55888-0_2
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55886-6
Online ISBN: 978-4-431-55888-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)