Advertisement

Methods for Inferences

  • Shuhei ManoEmail author
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

This chapter introduces inference methods based on the models presented in previous chapters. We discuss samplers, which are required for hypothesis testing and posterior sampling. After a brief introduction of Markov chain Monte Carlo (MCMC) samplers from A-hypergeometric distributions discussed in Chap.  3, we introduce a direct sampler, which allows us to draw independent samples directly from the target distribution. Gibbs partitions introduced in Chap.  2 and further discussed in Chap.  4 are related to A-hypergeometric distributions of two-rows matrices. We present some interesting topics on samplers from random partitions, including mixing assessment in terms of symmetric functions and construction of direct samplers by simulating stochastic processes on partitions. Finally, aided by information geometry, we discuss maximum likelihood estimation of curved exponential families, which arise in parameterization of the variables of A-hypergeometric distributions.

Keywords

A-hypergeometric distribution Curved exponential family Direct sampler Duality of Markov chains Gibbs partition Information geometry Markov chain Monte Carlo Mixing Symmetric function 

References

  1. 1.
    Diaconis, P., Sturmfels, B.: Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26, 363–397 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aoki, S., Hara, H., Takemura, A.: Markov Bases in Algebraic Statistics. Springer, New York (2012)CrossRefGoogle Scholar
  3. 3.
    Mano, S.: Partition structure and the A-hypergeometric distribution associated with the rational normal curve. Electron. J. Stat. 11, 4452–4487 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Green, P.J., Richardson, S.: Modelling heterogeneity with and without the Dirichlet process. Scand. J. Statist. 28, 355–375 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Study on the Japanese National Character. http://www.ism.ac.jp/kokuminsei/en/table/index.htm
  7. 7.
    Risa/Asir (Kobe distribution) Download Page. http://www.math.kobe-u.ac.jp/Asir/asir.html
  8. 8.
    Goto, Y., Tachibana, Y., Takayama, N.: Implementation of difference holonomic gradient methods for two-way contingency table. Comput. Algebra Relat. Top. RIMS Kôkyûroku 2054, 11 (2016)Google Scholar
  9. 9.
    Tachibana, Y., Goto, Y., Koyama, T., Takayama, N.: Holonomic gradient method for two way contingency tables. arXiv: 1803.04170
  10. 10.
    Diaconis, P., Eisenbud, B., Sturmfels, B.: Lattice walks and primary decomposition. In: Sagan, B.E., Stanley, R.P. (eds.) Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996). Progress in Mathematics, vol. 161, pp. 173–193. Birkhäuser, Boston (1998)CrossRefGoogle Scholar
  11. 11.
    Stewart, F.M.: Computer algorithm for obtaining a random set of allele frequencies for a locus in an equilibrium population. Genetics 86, 482–483 (1997)Google Scholar
  12. 12.
    Hanlon, P.: A Markov chain on the symmetric group and Jack symmetric functions. Discret. Math. 90, 123–140 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Diaconis, P., Lam, A.: A probabilistic interpretation of the Macdonald polynomials. Ann. Probab. 40, 1861–1896 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Newman, M.E.J., Barkema, G.T.: Monte Carlo Methods in Statistical Physics. Clarendon Press, New York (1999)zbMATHGoogle Scholar
  15. 15.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  16. 16.
    Tavaré, S: Ancestral inference in population genetics In: Ecole d’Été de Probabilités de Saint Flour, Lecture Notes in Math. vol. 1837. Springer, Berlin (2004)Google Scholar
  17. 17.
    Griffiths, R.C., Tavaré, S.: Ancestral inference in population genetics. Statist. Sci. 9, 307–319 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Forsythe, G.E., Leibler, R.A.: Matrix inversion by the Monte Carlo method. Math. Comp. 26, 127–129 (1950)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stephens, M., Donnelly, P.: Inference in molecular population genetics. J. R. Stat. Soc. Ser. B 62, 605–635 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    De Iorio, M., Griffiths, R.C.: Importance sampling on coalescent histories. I. Adv. Appl. Probab. 36, 417–433 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252 (1996)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fearnhead, P.: Perfect simulation from population genetic models with selection. Theor. Popul. Biol. 59, 263–279 (2001)CrossRefGoogle Scholar
  23. 23.
    Mano, S.: Ancestral graph with bias in gene conversion. J. Appl. Probab. 50, 239–255 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)CrossRefGoogle Scholar
  25. 25.
    Shimizu, A.: A measure valued diffusion process describing an \(n\) locus model incorporating gene conversion. Nagoya Math. J. 119, 81–92 (1990)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monograph vol. 191. Amer. Math. Soc. Providence (2000)Google Scholar
  27. 27.
    Levin, B., Reeds, J.: Compound multinomial likelihood functions are unimodal: proof of a conjecture of I.J. Good. Ann. Statist. 5, 79–87 (1977)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Keener, R., Rothman, E., Starr, N.: Distribution of partitions. Ann. Statist. 15, 1466–1481 (1978)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hoshino, N.: Applying Pitman’s sampling formula to microdata disclosure risk assessment. J. Official Statist. 17, 499–520 (2001)Google Scholar
  30. 30.
    Takayama, N., Kuriki, S., Takemura, A.: \(A\)-hypergeometric distributions and Newton polytopes. Adv. in Appl. Math. 99, 109–133 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Baayen, R.H.: Word Frequency Distribution. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan

Personalised recommendations