Statistics Surveys

Distributional properties of means of random probability measures

Antonio Lijoi and Igor Prünster

Full-text: Open access

Abstract

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.

Article information

Source
Statist. Surv., Volume 3 (2009), 47-95.

Dates
First available in Project Euclid: 11 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1249996755

Digital Object Identifier
doi:10.1214/09-SS041

Mathematical Reviews number (MathSciNet)
MR2529667

Zentralblatt MATH identifier
1190.62056

Subjects
Primary: 62F15: Bayesian inference 62E15: Exact distribution theory
Secondary: 60G57: Random measures

Keywords
Bayesian Nonparametrics completely random measures Cifarelli–Regazzini identity Dirichlet process functionals of random probability measures generalized Stieltjes transform neutral to the right processes normalized random measures posterior distribution random means random probability measure two–parameter Poisson–Dirichlet process

Citation

Lijoi, Antonio; Prünster, Igor. Distributional properties of means of random probability measures. Statist. Surv. 3 (2009), 47--95. doi:10.1214/09-SS041. https://projecteuclid.org/euclid.ssu/1249996755


Export citation

References

  • [1] Barlow, M., Pitman, J. and Yor, M. (1989). Une extension multidimensionnelle de la loi de l’arc sinus. In, Séminaire de Probabilités XXIII (Azema, J., Meyer, P.-A. and Yor, M., Eds.), 294–314, Lecture Notes in Mathematics 1372. Springer, Berlin.
  • [2] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes., Prob. Surveys 2, 191–212.
  • [3] Blackwell, D. (1973). Discreteness of Ferguson selections., Ann. Statist. 1, 356–358.
  • [4] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes., Adv. Appl. Probab. 31, 929–953.
  • [5] Carlton, M.A. (2002). A family of densities derived from the three-parameter Dirichlet process., J. Appl. Probab. 39, 764–774.
  • [6] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In, Exponential functionals and principal values related to Brownian motion (M. Yor, ed.), 73–126. Biblioteca de la Revista Matematica Ibero–Americana, Madrid.
  • [7] Cifarelli, D.M. and Melilli, E. (2000). Some new results for Dirichlet priors., Ann. Statist. 28, 1390–1413.
  • [8] Cifarelli, D.M. and Regazzini, E. (1979a). A general approach to Bayesian analysis of nonparametric problems. The associative mean values within the framework of the Dirichlet process. I. (Italian), Riv. Mat. Sci. Econom. Social. 2, 39–52.
  • [9] Cifarelli, D.M. and Regazzini, E. (1979b). A general approach to Bayesian analysis of nonparametric problems. The associative mean values within the framework of the Dirichlet process. II. (Italian), Riv. Mat. Sci. Econom. Social. 2, 95–111.
  • [10] Cifarelli, D.M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process., Ann. Statist., 18, 429–442 (Correction in Ann. Statist. (1994) 22, 1633–1634).
  • [11] Cifarelli, D.M. and Regazzini, E. (1993). Some remarks on the distribution functions of means of a Dirichlet process., Technical Report 93.4, IMATI–CNR, Milano.
  • [12] Cifarelli, D.M. and Regazzini, E. (1996). Tail-behaviour and finiteness of means of distributions chosen from a Dirichlet process., Technical Report 96.19, IMATI–CNR, Milano. Available at: http://www.mi.imati.cnr.it/iami/papers/96-19.ps
  • [13] Constantine, G.M. and Savits, T.H. (1996). A multivariate Faà di Bruno formula with applications., Trans. Amer. Math. Soc. 348, 503–520.
  • [14] Daley, D.J. and Vere–Jones, D. (1988)., An introduction to the theory of point processes. Springer, New York.
  • [15] Diaconis, P. and Kemperman, J. (1996). Some new tools on Dirichlet priors. In, Bayesian statistics 5 (J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, eds.) 97–106. Oxford Univ. Press.
  • [16] Dickey, J.M. and Jiang, T.J. (1998). Filtered-variate prior distributions for histogram smoothing., J. Amer. Statist. Assoc. 93, 651–662.
  • [17] Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions., Ann. Probab. 2, 183–201.
  • [18] Dykstra, R.L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability., Ann. Statist. 9, 356–367.
  • [19] Epifani, I., Guglielmi, A. and Melilli, E. (2006). A stochastic equation for the law of the random Dirichlet variance., Statist. Probab. Lett. 76, 495–502.
  • [20] Epifani, I., Lijoi, A. and Prünster, I. (2003). Exponential functionals and means of neutral–to–the–right priors., Biometrika 90, 791–808.
  • [21] Erhardsson, T. (2008). Nonparametric Bayesian inference for integrals with respect to an unknown finite Mmeasure., Scand. J. Statist. 35, 354–368.
  • [22] Evans, M.R. and Hanney, T. (2005). Nonequilibrium statistical mechanics of the zero-range process and related models., J. Phys. A 38, R195–R240.
  • [23] Exton, H. (1976)., Multiple hypergeometric functions and applications. Ellis Horwood, Chichester.
  • [24] Favaro, S., Guglielmi, A. and Walker, S.G. (2009). A class of measure-valued Markov Chains and Bayesian Nonparametrics., Tech. Report.
  • [25] Feigin, P.D. and Tweedie, R.L. (1989). Linear functionals and Markov chains associated with Dirichlet processes., Math. Proc. Camb. Phil. Soc. 105, 579–585.
  • [26] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems., Ann. Statist. 1, 209–230.
  • [27] Ferguson, T.S. (1974). Prior distributions on spaces of probability measures., Ann. Statist. 2, 615–629.
  • [28] Guglielmi, A. (1998). A simple procedure calculating the generalized Stieltjes transform of the mean of a Dirichlet process., Statist. Probab. Lett. 38, 299–303.
  • [29] Guglielmi, A. and Tweedie, R.L. (2001). Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process., Bernoulli 7, 573–592.
  • [30] Guglielmi, A., Holmes, C.C. and Walker, S.G. (2002). Perfect simulation involving functionals of a Dirichlet process., J. Comput. Graph. Statist. 11, 306–310.
  • [31] Gurland, J. (1948). Inversion formulae for the distributions of ratios., Ann. Math. Statist., 19, 228–237.
  • [32] Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2008). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models., Ann. Probab. 36, 1790–1837.
  • [33] Hannum, R.C., Hollander, M. and Langberg, N.A. (1981). Distributional results for random functionals of a Dirichlet process., Ann. Probab. 9, 665–670.
  • [34] Hill, T. and Monticino, M. (1998). Constructions of random distributions via sequential barycenters., Ann. Statist. 26, 1242–1253.
  • [35] Hjort, N.L. (2003). Topics in non-parametric Bayesian statistics. In:, Highly structured stochastic systems (P. Green, N.L. Hjort and S. Richardson, Eds.), 455–487. Oxford Univ. Press, Oxford.
  • [36] Hjort, N.L. and Ongaro, A. (2005). Exact inference for random Dirichlet means., Stat. Inference Stoch. Process. 8, 227–254.
  • [37] Hjort, N.L. and Petrone, S. (2007). Nonparametric quantile inference using Dirichlet processes. In, Advances in statistical modeling and inference, 463–492, World Sci. Publ., Hackensack, NJ.
  • [38] Ishwaran, H. and James, L.F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models., Statist. Sinica 13, 1211–1235.
  • [39] Ismail, M.E.H. and Kelker, D.H. (1979). Special functions, Stieltjes transforms and infinite divisibility., SIAM J. Math. Anal. 10, 884–901.
  • [40] James, L.F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics., Mathematics ArXiv, math.PR/0205093.
  • [41] James, L.F. (2006). Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and beta-gamma processes., Ann. Statist. 33, 647–660.
  • [42] James, L.F. (2007). New Dirichlet mean identities., Mathematics Arxiv, math.PR/0708.0614.
  • [43] James, L.F., Lijoi, A. and Prünster, I. (2006). Conjugacy as a distinctive feature of the Dirichlet process., Scand. J. Statist. 33, 105–120.
  • [44] 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.
  • [45] James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measures with independent increments., Scand. J. Statist. 36, 76–97.
  • [46] James, L.F., Lijoi, A. and Prünster, I. (2009). On the posterior distribution of classes of random means., Bernoulli, DOI: 10.3150/09-BEJ200.
  • [47] James, L.F., Roynette, B. and Yor, M. (2008). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples., Probab. Surv. 5, 346–415.
  • [48] Jarner, S.F., Tweedie, R.L. (2002). Convergence rates and moments of Markov chains associated with the mean of Dirichlet, processes.Stochastic Process. Appl. 101, 257–271.
  • [49] Kerov, S.V. (1998). Interlacing measures. In, Kirillov’s seminar on representation theory. Amer. Math. Soc. Transl. Ser. 2, Vol. 181. Amer. Math. Soc., Providence, 35–83.
  • [50] Kerov, S.V. and Tsilevich, N.V. (2001). The Markov-Krein correspondence in several dimensions., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283, 98–122. (English translation in J. Math. Sci. 121, 2345–2359).
  • [51] Kingman, J.F.C. (1967). Completely random measures., Pacific J. Math. 21, 59–78.
  • [52] Kingman, J.F.C. (1975). Random discrete distributions (with discussion)., J. Roy. Statist. Soc. Ser. B 37, 1–22.
  • [53] Kingman, J.F.C. (1993)., Poisson processes. Oxford University Press, Oxford.
  • [54] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes., Trans. Amer. Math. Soc. 88, 380–387.
  • [55] Lévy, P. (1939). Sur certains processus stochastiques homogènes., Compositio Math. 7, 283–339.
  • [56] Lijoi, A., Mena, R.H. and Prünster, I. (2005). Hierarchical mixture modelling with normalized inverse Gaussian priors., J. Amer. Statist. Assoc. 100, 1278–1291.
  • [57] Lijoi, A., Mena, R.H. and Prünster, I. (2005). Bayesian nonparametric analysis for a generalized Dirichlet process prior., Statist. Inference Stoch. Process., 8, 283–309.
  • [58] Lijoi, A., Mena, R.H. and Prünster, I. (2007). Controlling the reinforcement in Bayesian nonparametric mixture models., J. Roy. Statist. Soc. Ser. B 69, 715–740.
  • [59] Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions., Ann. Probab. 32, 1469–1495.
  • [60] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates., Ann. Statist. 12, 351–357.
  • [61] Lo, A.Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rate estimates., Ann. Inst. Statist. Math. 41, 227–245.
  • [62] Muliere, P. and Tardella, L. (1998). Approximating distributions of random functionals of Ferguson-Dirichlet priors., Canad. J. Statist. 26 283–297.
  • [63] Nieto-Barajas, L.E., Prünster, I. and Walker, S.G. (2004). Normalized random measures driven by increasing additive processes., Ann. Statist. 32, 2343–2360.
  • [64] Peccati, G. (2008). Multiple integral representation for functionals of Dirichlet processes., Bernoulli 14, 91–124.
  • [65] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions., Probab. Theory Related Fields. 102, 145–158.
  • [66] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In, Statistics, Probability and Game Theory (T.S. Ferguson, L.S. Shapley and J.B. MacQueen, Eds.). IMS Lecture Notes Monogr. Ser., Vol. 30. Inst. Math. Statist., Hayward, 245–267.
  • [67] Pitman, J. (2003). Poisson-Kingman partitions. In, Statistics and science: a Festschrift for Terry Speed (D.R Goldstein, Ed.). IMS Lecture Notes Monogr. Ser., Vol. 40. Inst. Math. Statist., Beachwood, 1–34.
  • [68] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator., Proc. London Math. Soc. 65 326–356.
  • [69] Pitman, J., and Yor, M. (1997). On the relative lengths of excursions derived from a stable subordinator. In:, Séminaire de Probabilités XXXI (Azema, J., Emery, M. and Yor, M., Eds.), 287–305, Lecture Notes in Mathematics 1655. Springer, Berlin.
  • [70] Pulkkinen, O. (2007). Boundary driven zero-range processes in random media., J. Stat. Phys. 128, 1289–1305.
  • [71] Regazzini, E. (1998). An example of the interplay between statistics and special functions. In, Atti dei convegni Lincei. Tricomi’s ideas and contemporary applied mathematics 147, 303–320.
  • [72] Regazzini, E., Guglielmi, A. and Di Nunno, G. (2002). Theory and numerical analysis for exact distribution of functionals of a Dirichlet process., Ann. Statist. 30, 1376–1411.
  • [73] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of random measures with independent increments., Ann. Statist. 31, 560–585.
  • [74] Regazzini, E. and Sazonov, V.V. (2001). Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference., Th. Prob. Appl., 45, 93–110.
  • [75] Romik, D. (2004). Explicit formulas for hook walks on continual Young diagrams., Adv. in Appl. Math. 32, 625–654.
  • [76] Romik, D. (2005). Roots of the derivative of a polynomial., Amer. Math. Monthly 112, 66–69.
  • [77] Sato, K. (1999)., Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge.
  • [78] Sethuraman, J. (1994). A constructive definition of the Dirichlet process prior., Statist. Sinica 2, 639–650.
  • [79] Schwarz, J.H. (2005) The generalized Stieltjes transform and its inverse., J. Math. Phys. 46 013501-8.
  • [80] Spitzer, F. (1970). Interaction of Markov processes., Advances in Math. 5, 246–290.
  • [81] Stanley, R.P. (1999)., Enumerative combinatorics. Volume 2. Cambridge University Press.
  • [82] Sumner, D.B. (1949). An inversion formula for the generalized Stieltjes transform., Bull. Amer. Math. Soc. 55, 174–183.
  • [83] Tamura, H. (1988). Estimation of rare errors using expert judgement., Biometrika 75, 1–9.
  • [84] Tsilevich, N.V. (1997). Distribution of mean values for some random measures., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 240, 268–279. (English translation in J. Math. Sci. 96, 3616–3623).
  • [85] Vershik, A.M., Yor, M. and Tsilevich, N.V. (2001). The Markov-Krein identity and the quasi-invariance of the gamma process., Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283, 21–36. (English translation in J. Math. Sci. 121, 2303–2310).
  • [86] Walker, S.G. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme., Ann. Statist. 25, 1762–1780.
  • [87] Yamato, H. (1980). On behaviors of means of distributions with Dirichlet processes., Rep. Fac. Sci. Kagoshima Univ. 13, 41–45.
  • [88] Yamato, H. (1984a). Expectations of functions of samples from distributions chosen from Dirichlet processes., Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. 17, 1–8.
  • [89] Yamato, H. (1984b). Characteristic functions of means of distributions chosen from a Dirichlet process., Ann. Probab. 12, 262–267.
  • [90] Yano, K. and Yano, Y. (2008). Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions., Statist. Prob. Letters 78, 2175–2180.