Electronic Journal of Statistics

Projective limit random probabilities on Polish spaces

Peter Orbanz

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A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals—the most prominent example being the construction of the Dirichlet process from finite-dimensional Dirichlet distributions. This approach is both intuitive and applicable to the construction of arbitrary distributions on M(V), but also hamstrung by a number of technical difficulties. We show how these difficulties can be resolved if the domain V is a Polish topological space, and give a representation theorem directly applicable to the construction of any probability distribution on M(V) whose first moment measure is well-defined. The proof draws on a projective limit theorem of Bochner, and on properties of set functions on Polish spaces to establish countable additivity of the resulting random probabilities.

Article information

Electron. J. Statist., Volume 5 (2011), 1354-1373.

First available in Project Euclid: 19 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 60G57: Random measures

Bayesian nonparametrics Dirichlet processes random probability measures


Orbanz, Peter. Projective limit random probabilities on Polish spaces. Electron. J. Statist. 5 (2011), 1354--1373. doi:10.1214/11-EJS641. https://projecteuclid.org/euclid.ejs/1319028571

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  • [1] Aliprantis, C. D. and Border, K. C. (2006)., Infinite Dimensional Analysis . Springer, 3rd edition.
  • [2] Bauer, H. (1996)., Probability Theory . W. de Gruyter.
  • [3] Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes., Ann. Statist., 1, 353–355.
  • [4] Bochner, S. (1955)., Harmonic Analysis and the Theory of Probability . University of California Press.
  • [5] Bourbaki, N. (1966)., Elements of Mathematics: General Topology . Hermann (Paris) and Addison-Wesley.
  • [6] Bourbaki, N. (2004)., Elements of Mathematics: Integration . Springer.
  • [7] Crauel, H. (2002)., Random probability measures on Polish spaces . Taylor & Francis.
  • [8] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems., Ann. Statist., 1(2).
  • [9] Ferguson, T. S. (1974). Prior distributions on spaces of probability measures., Ann. Statist., 2(4), 615–629.
  • [10] Fremlin, D. H. (2000–2006)., Measure Theory , volume I–IV. Torres Fremlin.
  • [11] Gaudard, M. and Hadwin, D. (1989). Sigma-algebras on spaces of probability measures., Scand. J. Stat., 16, 169–165.
  • [12] Ghosal, S. (2010). Dirichlet process, related priors and posterior asymptotics. In N. L. Hjort, et al., editors, Bayesian Nonparametrics . Cambridge University Press.
  • [13] Ghosh, J. K. and Ramamoorthi, R. V. (2002)., Bayesian Nonparametrics . Springer.
  • [14] Harris, T. E. (1968). Counting measures, monotone random set functions., Probab. Theory Related Fields , 10, 102–119.
  • [15] Kallenberg, O. (1983)., Random Measures . Academic Press.
  • [16] Kallenberg, O. (2001)., Foundations of Modern Probability . Springer, 2nd edition.
  • [17] Kechris, A. S. (1995)., Classical Descriptive Set Theory . Springer.
  • [18] Kingman, J. F. C. (1975). Random discrete distributions., J. R. Stat. Soc. Ser. B Stat. Methodol., 37, 1–22.
  • [19] Lavine, M. (1992). Some aspects of Pólya tree distributions for statistical modelling., Ann. Statist., 20(3), 1222–1235.
  • [20] Lijoi, A., Mena, R. H., and Prünster, I. (2005). Hierarchical mixture modeling with normalized inverse-Gaussian priors., J. Amer. Statist. Assoc., 100, 1278–1291.
  • [21] MacEachern, S. N. (2000). Dependent Dirichlet processes. Technical report, Ohio State, University.
  • [22] Mallory, D. J. and Sion, M. (1971). Limits of inverse systems of measures., Ann. Inst. Fourier (Grenoble) , 21(1), 25–57.
  • [23] Olshanski, G. (2003). An introduction to harmonic analysis on the infinite symmetric group. In, Asymptotic Combinatorics with Applications to Mathematical Physics , volume 1815 of Lecture Notes in Mathematics , pages 127–160. Springer.
  • [24] Pollard, D. (1984)., Convergence of Stochastic Processes .
  • [25] Sethuraman, J. (1994). A constructive definition of Dirichlet priors., Statist. Sinica , 4, 639–650.
  • [26] Talagrand, M. (2003)., Spin Glasses: A Challenge for Mathematicians . Springer.
  • [27] Walker, S. G., Damien, P., Laud, P. W., and Smith, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions., J. R. Stat. Soc. Ser. B Stat. Methodol., 61(3), 485–527.
  • [28] Zhao, L. H. (2000). Bayesian aspects of some nonparametric problems., Ann. Statist., 28, 532–552.