## Electronic Journal of Statistics

### Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling

Annalisa Cerquetti

#### Abstract

Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Here we call into question the current approach to Bayesian nonparametric estimation in species sampling problems under Gibbs priors, which incorrectly relies on treating exchangeable partition probability functions (EPPFs) as multivariate distributions on compositions of the positive integers. We show that once those multivariate distributions are correctly derived, results for corresponding sampling formulas can be obtained, generalized and sometimes fixed, working with marginals and a known result on falling factorial moments of a sum of non independent indicators. We provide an application of our findings to a recently proposed Bayesian nonparametric estimation under Gibbs priors of the predictive probability to observe a species already observed a certain number of times.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 697-716.

Dates
First available in Project Euclid: 17 March 2013

https://projecteuclid.org/euclid.ejs/1363482367

Digital Object Identifier
doi:10.1214/13-EJS784

Mathematical Reviews number (MathSciNet)
MR3035269

Zentralblatt MATH identifier
1327.62196

Subjects
Primary: 60G57: Random measures 62G05: Estimation
Secondary: 62F15: Bayesian inference

#### Citation

Cerquetti, Annalisa. Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling. Electron. J. Statist. 7 (2013), 697--716. doi:10.1214/13-EJS784. https://projecteuclid.org/euclid.ejs/1363482367

#### References

• [1] Arratia, R., Barbour, A. D., Tavaré, S. (2003), Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics.
• [2] Cerquetti, A. (2011a) A decomposition approach to Bayesian nonparametric estimation under two-parameter Poisson-Dirichlet priors., Proceedings of ASMDA 2011 - Rome, Italy. Available at: http://geostasto.eco.uniroma1.it/utenti/cerquetti/asmda2011last.pdf
• [3] Cerquetti, A. (2011b) Conditional $\alpha$-diversity for exchangeable Gibbs partition driven by the stable subordinator., Proceedings of S.Co. Conferemce, 2011, Padova, Italy. Available at: http://homes.stat.unipd.it/mgri/SCo2011/Papers/CS/CS-7/cerquetti.pdf.
• [4] Charalambides, C. A. (2005), Combinatorial Methods in Discrete Distributions. Wiley, Hoboken NJ.
• [5] De Moivre, A. (1718), The doctrine of chances: Or a method of calculating the probabilities of events in play. London. Pearson.
• [6] Ewens, W. J. (1972) The sampling theory of selectively neutral alleles., Theoret. Pop. Biol., 3, 87–112.
• [7] Ewens, W. and Tavaré, S. (1995) The Ewens sampling formula. In Multivariate discrete distributions (Johnson, N.S., Kotz, S. and Balakrishnan, N. eds.). Wiley, NY.
• [8] Favaro, S., Lijoi, A., Mena, R. H., Prünster, I. (2009) Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior., J. Roy. Statist. Soc. B, 71, 993–1008.
• [9] Favaro, S., Lijoi, A. and Prünster, I. (2012a) Conditional formulae for Gibbs-type exchangeable random partitions., Ann. Appl. Probab. (to appear)
• [10] Favaro, S., Lijoi, A. and Prünster, I. (2012b) A new estimator of the discovery probability., Biometrics, 68, 1188–1196.
• [11] Ferguson, T. S. (1973) A Bayesian analysis of some nonparametric problems., Ann. Statist., 1, 209–230.
• [12] Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943) 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.
• [13] Gnedin, A. (2010) A species sampling model with finitely many types., Electron. Commun. Prob., 15, 79–88.
• [14] Gnedin, A. and Pitman, J. (2006) Exchangeable Gibbs partitions and Stirling triangles., J. Math. Sci., 138, 3, 5674–5685.
• [15] Hsu, L. C, and Shiue, P. J. (1998) A unified approach to generalized Stirling numbers., Adv. Appl. Math., 20, 366-384.
• [16] Iyer, P. V. K. (1949) Calculation of factorial moments of certain probability distributions., Nature. 164, 282.
• [17] Iyer, P. V. K. (1958) A theorem on factorial moments and its applications., Ann. Math. Statist., 29, 254–261.
• [18] Johnson, N. S. and Kotz, S. (2005), Univariate discrete distributions 3rd Ed. Wiley, NY.
• [19] Jordan, M. C. (1867) De quelques formules de probabilité., Comptes Rendus. Académie des Sciences, Paris, 65, 993–994.
• [20] Kingman, J. F. C. (1975) Random discrete distributions., J. Roy. Statist. Soc. B, 37, 1–22.
• [21] Kingman, J. F. C (1978) The representation of partition structure., J. London Math. Soc. 2, 374–380.
• [22] Lijoi, A., Mena, R. and Prünster, I. (2007) Bayesian nonparametric estimation of the probability of discovering new species., Biometrika, 94, 769–786.
• [23] Lijoi, A., Prünster, I. and Walker, S. G. (2008) Bayesian nonparametric estimators derived from conditional Gibbs structures., Ann. Appl. Probab., 18, 1519–1547.
• [24] Normand, J. M. (2004) Calculation of some determinants using the $s$-shifted factorial., J. Phys. A: Math. Gen. 37, 5737-5762.
• [25] Pitman, J. (1995) Exchangeable and partially exchangeable random partitions., Probab. Th. Rel. Fields, 102, 145-158.
• [26] Pitman, J. (1996) Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson, Shapley L.S., and MacQueen J.B., editors, Statistics, Probability and Game Theory, vol. 30 of IMS Lecture Notes-Monograph Series, pages 245–267. Institute of Mathematical Statistics, Hayward, CA.
• [27] Pitman, J. (2003) Poisson-Kingman partitions. In D.R. Goldstein, editor, Science and Statistics: A Festschrift for Terry Speed, volume 40 of Lecture Notes-Monograph Series, pages 1–34. IMS, Hayward, California.
• [28] Pitman, J. (2006), Combinatorial Stochastic Processes. Ecole d’Eté de Probabilité de Saint-Flour XXXII - 2002. Lecture Notes in Mathematics N. 1875, Springer.
• [29] Pitman, J. and Yor, M. (1997) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator., Ann. Probab., 25, 855–900.
• [30] Yamato, H. and Sibuya, M. (2000) Moments of some statistics of Pitman sampling formula., Bull. Inform. Cybernet., 32, 1.