Bayesian Estimation of Gini-Simpson’s Index Under Mainland-Island Community Structure

  • Annalisa CerquettiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 274)


The mainland-island community structure is an ecological transposition of a popular model in population genetics in which a fixed number of subpopulations (islands) are connected, through differing immigration rates, to a single metapopulation (mainland) where diversity is generated through speciation. It has been recently shown that a large class of neutral models with this particular structure converges in the large population limit to the Hierarchical Dirichlet process. This finding provides the analogous, in the multipopulation setting, of the Ewens sampling formula for the single population neutral hypothesis. Here we apply some recent results for conditional moments of diversity measures under Gibbs-type priors to derive a Bayesian nonparametric estimator of Gini-Simpson’s index under the Hubbell Unified Neutral Theory of Biodiversity and Biogeography. Potential applications are also illustrated.


Bayesian estimator Gini Simpson index Diversity 


  1. 1.
    Cerquetti, A.: Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling. Electr. J. Stat. 169, 321–354 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cerquetti, A.: Bayesian nonparametric estimation of Tsallis diversity indices under Gnedin-Pitman priors (2014). arXiv:1404.3441v2 [math.ST]
  3. 3.
    Chao, A., Jost, L., Hsieh, T.C., Ma, K.H., Sherwin, W.B., Rollins, L.A.: Expected Shannon entropy and Shannon differentiation between subpopulations for neutral genes under the finite island model. Plos One 10, 1–2 (2015)Google Scholar
  4. 4.
    Etienne, R.S.: A new sampling formula for neutral biodiversity. Ecol. Lett. 7, 321–354 (2005)Google Scholar
  5. 5.
    Etienne, R.S.: A neutral sampling formula for multiple samples and an ’exact’ test of neutrality. Ecol. Lett. 10, 608–618 (2007)CrossRefGoogle Scholar
  6. 6.
    Etienne, R.S.: Maximum likelihood estimation of neutral model parameters for multiple samples with different degrees of dispersal limitation. J. Theor. Biol. 257, 510–514 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Etienne, R.S., Olff, H.: A novel genealogical approach to neutral biodiversity. Ecol. Lett. 8, 253–260 (2004)CrossRefGoogle Scholar
  8. 8.
    Ewens, W.J.: The sampling theory of selectively neutral alleles. Theoret. Popul. Biol. 3, 87–112 (1972)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Favaro, S., Lijoi, A., Mena, R.H., Prünster, I.: Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior. J. R. Stat. Soc. B. 71, 993–1008 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Favaro, S., Lijoi, A., Prünster, I.: Conditional formulae for Gibbs-type exchangeable random partitions. Ann. Appl. Probab. 23, 1721–1754 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gini, C.W.: Variabilita e mutabilita. Studi Economico-Giuridici della R. Università di. Cagliari 3, 3–159 (1912)Google Scholar
  13. 13.
    Gnedin, A., Pitman, J.: Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. 138(3), 5674–5685 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Harris, K., Parsons, T. L., Ijaz, U. Z., Lahti, L., Holmes, I., Quince, C. Linking statistical and ecological theory: Hubbell’s unified neutral theory of biodiversity as a hierarchical Dirichlet process. Proceedings of the IEEE, Issue: 99(2015)Google Scholar
  15. 15.
    Hubbell, S.P.: The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton (2001)Google Scholar
  16. 16.
    Jabot, F., Chave, J.: Analyzing tropical forest tree species abundance distributions using a nonneutral model and through approximate Bayesian inference. Am. Nat. 178(2), 38–47 (2011)CrossRefGoogle Scholar
  17. 17.
    Jost, L.: Partitioning diversity into independent alpha and beta components. Ecology 88, 2427–2439 (2007)CrossRefGoogle Scholar
  18. 18.
    Kimura, M.: Evolutionary rate at the molecular level. Nature 217, 624–626 (1968)CrossRefGoogle Scholar
  19. 19.
    Kingman, J.F.C.: Random discrete distributions. J. R. Stat. Soc. B 37, 1–22 (1975)MathSciNetzbMATHGoogle Scholar
  20. 20.
    MacArthur, R.H., Wilson, E.O.: The Theory of Island Biogeography. Princeton University Press, Princeton (1967)Google Scholar
  21. 21.
    Patil, G.P., Taille, C.: Diversity as a concept and its measurement. J. Am. Stat. Assoc. 77, 548–561 (1982)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pitman, J.: Poisson-Kingman partitions. In: Goldstein, D.R. (ed.) Science and Statistics: A Festschrift for Terry Speed. LN Monograph Series, vol. 40, pp. 1–34. IMS, Hayward (2003)Google Scholar
  23. 23.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Simpson, E.H.: Measurement of diversity. Nature 163, 688 (1949)CrossRefGoogle Scholar
  25. 25.
    Teh, Y.W., Jordan, M.I., Beal, M.J., Blei, D.M.: Hierarchical Dirichlet processes. J. Am. Stat. Assoc. 101, 1566–1581 (2006)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Watterson, G.A.: Heterosis or neutrality? Genetics 88, 405–417 (1977)MathSciNetGoogle Scholar
  28. 28.
    Xing, E.P., Sohn, K., Jordan, M.I., Teh, Y.W.: Bayesian Multi-Population Haplotype Inference via a Hierarchical Dirichlet Process Mixture. Proceedings of the 23rd International Conference on Machine Learning (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.MEMOTEF, Sapienza Università di RomaRomaItaly

Personalised recommendations