Electronic Journal of Statistics

On the stick-breaking representation of $\sigma$-stable Poisson-Kingman models

Stefano Favaro, Maria Lomeli, Bernardo Nipoti, and Yee Whye Teh

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Abstract

In this paper we investigate the stick-breaking representation for the class of $\sigma$-stable Poisson-Kingman models, also known as Gibbs-type random probability measures. This class includes as special cases most of the discrete priors commonly used in Bayesian nonparametrics, such as the two parameter Poisson-Dirichlet process and the normalized generalized Gamma process. Under the assumption $\sigma=u/v$, for any coprime integers $1\leq u<v$ such that $u/v\leq1/2$, we show that a $\sigma$-stable Poisson-Kingman model admits an explicit stick-breaking representation in terms of random variables which are obtained by suitably transforming Gamma random variables and products of independent Beta and Gamma random variables.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 1063-1085.

Dates
First available in Project Euclid: 5 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1407243242

Digital Object Identifier
doi:10.1214/14-EJS921

Mathematical Reviews number (MathSciNet)
MR3263112

Zentralblatt MATH identifier
1298.62049

Subjects
Primary: 62F15: Bayesian inference 60G57: Random measures

Keywords
Bayesian nonparametrics Beta random variable exponential tilting $G$-functions Gamma random variable discrete random probability measure $\sigma$-stable Poisson-Kingman model stick-breaking prior

Citation

Favaro, Stefano; Lomeli, Maria; Nipoti, Bernardo; Teh, Yee Whye. On the stick-breaking representation of $\sigma$-stable Poisson-Kingman models. Electron. J. Statist. 8 (2014), no. 1, 1063--1085. doi:10.1214/14-EJS921. https://projecteuclid.org/euclid.ejs/1407243242


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References

  • [1] Chaumont, L. and Yor, M. (2012). Exercises in Probability. A Guided Tour from Measure Theory to Random Processes, via Conditioning. Cambridge University Press, Cambridge.
  • [2] De Blasi, P., Favaro, S., Lijoi, A., Mena, R.H., Prünster, I. and Ruggiero, M. (2014). Are Gibbs-type priors the most natural generalization of the Dirichlet process? IEEE Transactions on Pattern Analysis and Machine Ingelligence, in press.
  • [3] De Iorio, M., Müller, P., Rosner, G. and MacEachern, S.N. (2004). An ANOVA model for dependent random measures. J. Amer. Statist. Assoc., 99, 205–215.
  • [4] Duan, J.A., Guindani, M. and Gelfand, A.E. (2007). Generalized spatial Dirichlet process models. Biometrika, 94, 809–825.
  • [5] Dunson, D.B. and Park, J.H. (2008). Kernel stick-breaking processes. Biometrika, 95, 307–323.
  • [6] Dunson, D.B., Xue, Y. and Carin, L. (2008). The matrix stick-breaking process: flexible Bayes meta analysis. J. Amer. Statist. Assoc., 103, 317–327.
  • [7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1976). Higher Trascendental Functions Vol. I. McGraw-Hill, New York.
  • [8] Favaro, S., Lijoi, A. and Prünster, I. (2012). On the stick-breaking representation of normalized inverse Gaussian priors. Biometrika, 99, 663–674.
  • [9] Favaro, S. and Walker S.G. (2013). Slice sampling $\sigma$-stable Poisson-Kingman mixture models. J. Comput. Graph. Statist., 22, 830–847.
  • [10] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209–230.
  • [11] Gnedin, A. and Pitman, J. (2005). Exchangeable Gibbs partitions and Stirling triangles. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 325, 83–102.
  • [12] Gradshteyn, L.S. and Ryzhik, L.M. (2000). Table of Integrals, Series, and Products. Academic Press, New York.
  • [13] Griffin, J. and Steel, M.F.J. (2006). Order-based dependent Dirichlet processes. J. Amer. Stat. Assoc., 101, 179–194.
  • [14] Hofert, M. (2011). Efficiently sampling nested Archimedean copulas. Comput. Statist. Data Anal., 55, 57–70.
  • [15] Ishwaran, H. and James, L.F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Stat. Ass., 96, 161–173.
  • [16] James, L.F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Preprint arXiv:math/0205093.
  • [17] James, L.F., Lijoi, A. and Prünster, I. (2008). Distributions of linear functionals of two parameter Poisson-Dirichlet random measures. Ann. Appl. Probab., 18, 521–551.
  • [18] James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measures with independent increments. Scand. J. Statist., 36, 76–97.
  • [19] James, L.F. (2013). Stick-breaking PG$(\alpha,\zeta)$-generalized Gamma processes. Preprint arXiv:1308.6570.
  • [20] Kingman, J.F.C (1967). Completely random measures. Pacific J. Math., 21, 59–78.
  • [21] Kingman, J.F.C (1975). Random discrete distributions. J. Roy. Statist. Soc., Series B, 37, 1–22.
  • [22] Lijoi, A., Mena, R.H. and Prünster, I. (2005). Hierarchical mixture modelling with normalized inverse Gaussian priors. J. Amer. Stat. Assoc., 100, 1278–1291.
  • [23] Lijoi, A., Mena, R.H. and Prünster, I. (2007). Controlling the reinforcement in Bayesian nonparametric mixture models. J. Roy. Statist. Soc., Series B, 69, 715–740.
  • [24] Lijoi, A., Prünster, I. and Walker, S.G. (2008). Investigating nonparametric priors with Gibbs structure. Statist. Sinica, 18, 1653–1668.
  • [25] MacEachern, S.N. (1999). Dependent nonparametric processes. In ASA Proceedings of the Section on Bayesian Statistical Science, Alexandria, VA: American Statistical Association.
  • [26] MacEachern, S.N. (2000). Dependent Dirichlet processes. Technical Report, Ohio State University.
  • [27] Meijer, G.S. (1936). Uber Whittakersche bezw. Besselsche funktionen und deren produkte. Nieuw. Arch. Wiskunde, 18, 10–39.
  • [28] Muliere, P. and Tardella, L. (1998). Approximating distributions of random functionals of Ferguson-Dirichlet priors. Canad. J. Statist., 26, 283–298.
  • [29] Papaspiliopoulos, O. and Roberts, G.O. (2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models, Biometrika, 95, 169–186.
  • [30] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields, 92, 21–39.
  • [31] Petrone, S., Guindani, M. and Gelfand, A.E. (2009). Hybrid Dirichlet mixture models for functional data. J. Roy. Statist. Soc., Series B, 71, 755–782.
  • [32] Pitman, J. (1996). Random discrete distribution invariant under sized–biased permutation. Adv. Appl. Probab., 28, 525–539.
  • [33] Pitman, J. and Yor, M. (1997). The two parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 27, 855–900.
  • [34] Pitman, J. (2003). Poisson-Kingman partitions. In: Goldstein, D.R. (Ed.), Science and Statistics: A Festschrift for Terry Speed, Lecture Notes Monograph Series 40, 1–34. IMS, Beachwood, OH.
  • [35] Pitman, J. (2006). Combinatorial Stochastic Processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII. Lecture Notes in Mathematics 1875. Springer, New York.
  • [36] Prudnikov, A.P., Brychkov, Y.A. and Marichev, O.I. (1990) Integrals and Series Vol. III. Gordon and Breach Science Publishers, New York.
  • [37] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist., 31, 560–585.
  • [38] Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statist. Sinica, 4, 639–650.
  • [39] Sethuraman, J. and Tiwari, T.C. (1982). Convergence of Dirichlet measures and the interpretation of their parameter. In: Gupta, S.S. and Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics III.
  • [40] Springer, M.D. and Thompson, W.E. (1970). The distribution of products of Beta, Gamma and Gaussian random variables. SIAM J. Appl. Math., 18, 721–737.
  • [41] Springer, M.D. (2012). The Algebra of Random Variables. John Wiley and Sons.
  • [42] Teh, Y.W., Jordan, M.I., Beal, M.J. and Blei, D.M. (2006). Hierarchical Dirichlet Processes. J. Amer. Stat. Assoc., 101, 1566–1581.
  • [43] Walker, S.G. (2007). Sampling the Dirichlet mixture model with slices. Comm. Statist. Simul. Computat., 36, 45–54.
  • [44] Zolotarev, V.M. (1994). On representation of densities of stable laws by special functions. Theory Probab. Appl., 39, 354–362.

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