Polynomial time approximate or perfect samplers for discretized Dirichlet distribution

  • Tomomi Matsui
  • Mitsuo Motoki
  • Naoyuki Kamatani
  • Shuji KijimaEmail author
Original Paper Area 2


In this paper, we propose two Markov chains for sampling random vectors that are distributed according to discretized Dirichlet distribution. Their mixing rates are bounded by O(n 2 log Δ) and O(n 3 log Δ), where n is the dimension and 1/Δ is the grid size for discretization. Our second chain gives a perfect sampler that is based on monotone coupling from the past. We also report the results of simulations.


Markov chains Dirichlet distribution Path coupling Coupling from the past Perfect simulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: 38th Annual Symposium on Foundations of Computer Science, pp. 223–231. IEEE, San Alimitos (1997)Google Scholar
  2. 2.
    Bubley R.: Randomized Algorithms: Approximation, Generation, and Counting. Springer, New York (2001)zbMATHGoogle Scholar
  3. 3.
    Burr T.L.: Quasi-equilibrium theory for the distribution of rare alleles in a subdivided population: justification and implications. Theor. Popul. Biol. 57, 297–306 (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    Burr D., Doss H., Cooke G.E., Goldschmidt-Clermont P.J.: A meta-analysis of studies on the association of the platelet PlA polymorphism of glycoprotein IIIa and risk of coronary heart disease. Stat. Med. 22, 1741–1760 (2003)CrossRefGoogle Scholar
  5. 5.
    Chiang J., Chib S., Narasimhan C.: Markov chain Monte Carlo and models of consideration set and parameter heterogeneity. J. Econom. 89, 223–248 (1999)zbMATHCrossRefGoogle Scholar
  6. 6.
    Dyer M., Greenhill C.: Polynomial-time counting and sampling of two-rowed contingency tables. Theor. Comput. Sci. 246, 265–278 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dimakos X.K.: A guide to exact simulation. Int. Stat. Rev. 69, 27–48 (2001)zbMATHCrossRefGoogle Scholar
  8. 8.
    Durbin R., Eddy R., Krogh A., Mitchison G.: Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, London (1998)zbMATHGoogle Scholar
  9. 9.
    Fill J.: An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8, 131–162 (1988)MathSciNetGoogle Scholar
  10. 10.
    Fill J., Machida M., Murdoch D., Rosenthal J.: Extension of Fill’s perfect rejection sampling algorithm to general chains. Random Struct. Algorithms 17, 290–316 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Graham J., Curran J., Weir B.S.: Conditional genotypic probabilities for microsatellite loci. Genetics 155, 1973–1980 (2000)Google Scholar
  12. 12.
    Kijima S., Matsui T.: Polynomial time perfect sampling algorithm for two-rowed contingency tables. Random Struct. Algorithms 29, 243–256 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kijima, S., Matsui, T.: Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex. In: Proceedings of the 2005 International Conference on the Analysis of Algorithms (AofA). Discrete Mathematics and Computer Science, DMTCS Proceedings Series, vol. AD, pp. 369–380 (2005)Google Scholar
  14. 14.
    Kitada S., Hayashi T., Kishino H.: Empirical Bayes procedure for estimating genetic distance between populations and effective population size. Genetics 156, 2063–2079 (2000)Google Scholar
  15. 15.
    Laval G., SanCristobal M., Chevalet C.: Maximum-likelihood and Markov chain Monte Carlo approaches to estimate inbreeding and effective size form allele frequency changes. Genetics 164, 1189–1204 (2003)Google Scholar
  16. 16.
    Martin R., Randall D.: Disjoint decomposition of Markov chains and sampling circuits in Cayley graphs. Comb. Probab. Comput. 15, 411–448 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Matsui, T., Motoki, M., Kamatani, N.: Polynomial time approximate sampler for discretized Dirichlet distribution. In: Proceedings of 14th International Symposium on Algorithms and Computation (ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 676–685. Springer, Berlin (2003)Google Scholar
  18. 18.
    Matsui T., Kijima S.: Polynomial time perfect sampler for discretized Dirichlet distribution. In: Tsubaki, H., Nishina, K., Yamada, S. (eds) The Grammar of Technology Development, pp. 179–199. Springer, Berlin (2008)CrossRefGoogle Scholar
  19. 19.
    Matsui T., Matsui Y., Ono Y.: Random generation of 2 × 2 × . . . × 2 × J contingency tables. Theor. Comput. Sci. 326, 117–135 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
  21. 21.
    Niu T., Qin Z.S., Xu X., Liu J.S.: Bayesian haplotype inference for multiple linked single-nucleotide polymorphisms. Am. J. Hum. Genet. 70, 157–169 (2002)CrossRefGoogle Scholar
  22. 22.
    Pritchard J.K., Stephens M., Donnely P.: Inference of population structure using multilocus genotype data. Genetics 155, 945–959 (2000)Google Scholar
  23. 23.
    Propp J., Wilson D.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Propp J., Wilson D.: How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms 27, 170–217 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Randall, D., Winkler, P.: Mixing points on an interval. In: Proceedings of the Second Workshop on Analytic Algorithms and Combinatorics, pp. 216–221, Vancouver (2005)Google Scholar
  26. 26.
    Robert C.P.: The Bayesian Choice. Springer, New York (2001)zbMATHGoogle Scholar
  27. 27.
    Wilson D.: How to couple from the past using a read-once source of randomness. Random Struct. Algorithms 16, 85–113 (2000)zbMATHCrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer 2010

Authors and Affiliations

  • Tomomi Matsui
    • 1
  • Mitsuo Motoki
    • 2
  • Naoyuki Kamatani
    • 3
  • Shuji Kijima
    • 4
    Email author
  1. 1.Department of Information and System Engineering, Faculty of Science and EngineeringChuo UniversityTokyoJapan
  2. 2.Department of Computer Engineering and International CommunicationKanazawa Technical CollegeKanazawa, IshikawaJapan
  3. 3.Institute of RheumatologyTokyo Women’s Medical UniversityTokyoJapan
  4. 4.Department of InformaticsGraduate School of ISEE, Kyushu UniversityFukuokaJapan

Personalised recommendations