• Bernoulli
  • Volume 6, Number 6 (2000), 1021-1034.

A large-deviation principle for Dirichlet posteriors

Ayalvadi J. Ganesh and Neil O'connell

Full-text: Open access


Let Xk be a sequence of independent and identically distributed random variables taking values in a compact metric space Ω, and consider the problem of estimating the law of X1 in a Bayesian framework. A conjugate family of priors for nonparametric Bayesian inference is the Dirichlet process priors popularized by Ferguson. We prove that if the prior distribution is Dirichlet, then the sequence of posterior distributions satisfies a large-deviation principle, and give an explicit expression for the rate function. As an application, we obtain an asymptotic formula for the predictive probability of ruin in the classical gambler's ruin problem.

Article information

Bernoulli, Volume 6, Number 6 (2000), 1021-1034.

First available in Project Euclid: 5 April 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotics Bayesian nonparametrics Dirichlet process large deviations


Ganesh, Ayalvadi J.; O'connell, Neil. A large-deviation principle for Dirichlet posteriors. Bernoulli 6 (2000), no. 6, 1021--1034.

Export citation


  • [1] Barron, A., Schervish, M.J. and Wasserman, L. (1999) The consistency of posterior distributions in nonparametric problems. Ann. Statist., 27, 536-561. Abstract can also be found in the ISI/STMA publication
  • [2] Dembo, A. and Zeitouni, O. (1993) Large Deviations Techniques and Applications. Boston and London: Jones and Bartlett.
  • [3] Dupuis, P. and Ellis, R.S. (1997) A Weak Convergence Approach to the Theory of Large Deviations. New York: Wiley.
  • [4] Ferguson, T.S. (1973) A Bayesian analysis of some non-parametric problems. Ann. Statist., 1, 209- 230.
  • [5] Ferguson, T.S. (1974) Prior distributions on spaces of probability measures. Ann. Statist., 2, 615-629.
  • [6] Freedman, D. (1963) On the asymptotic behavior of Bayes estimates in the discrete case. Ann. Math. Statist., 34, 1386-1403.
  • [7] Ganesh, A.J. and O'Connell, N. (1999) An inverse of Sanov's theorem. Statist. Probab. Lett., 42, 201- 206.
  • [8] Ganesh, A., Green, P., O'Connell, N. and Pitts, S. (1998) Bayesian network management, Queueing Systems, 28, 267-282.
  • [9] Georgii, H.-O. (1988) Gibbs Measures and Phase Transitions. Berlin: De Gruyter.
  • [10] Ghosal, S., Ghosh, J.K. and van der Vaart, A.W. (1998) Convergence rates of posterior distributions. Preprint. Abstract can also be found in the ISI/STMA publication
  • [11] Ghosal, S., Ghosh, J.K. and Ramamoorthi, R.V. (1999) Consistency issues in Bayesian nonparametrics. In S. Ghosh (ed.), Asymptotics, Nonparametrics and Time Series. New York: Marcel Dekker.
  • [12] Lavine, M. (1992) Some aspects of Pólya tree distributions for statistical modelling. Ann. Statist., 20, 1222-1235.
  • [13] Lynch, J. and Sethuraman, J. (1987) Large deviations for processes with independent increments. Ann. Probab., 15, 610-627.
  • [14] Mauldin, R.D., Sudderth, W.D. and Williams, S.C. (1992) Pólya trees and random distributions. Ann. Statist., 20, 1203-1221.
  • [15] Rockafellar, R.T. (1974) Conjugate Duality and Optimization. Philadelphia: Society for Industrial and Applied Mathematics.
  • [16] Schwartz, L. (1965) On Bayes procedures. Z. Wahrscheinlichkeitstheorie. Verw. Geb., 4, 10-26.
  • [17] Shen, X. and Wasserman, L. (1998) Rates of convergence of posterior distributions. Technical report, Dept. of Statistics, Carnegie-Mellon University. Abstract can also be found in the ISI/STMA publication
  • [18] Wasserman, L. (1998) Asymptotic properties of nonparametric Bayesian procedures. Technical report, Dept. of Statistics, Carnegie-Mellon University.