Bayesian Nonparametrics (see Bayesian Nonparametric Statistics) took off with two papers of Ferguson (Ferguson 1974, 1983) and followed by Antoniak (Antoniak 1974). However consistency or asymptotics were not major issue in those papers, which were more concerned with taking the first steps towards a usable, easy to interpret prior with easy to choose hyperparameters and a rich support. Unfortunately, the fact that the Dirichlet sits on discrete distributions diminished the early enthusiasm.
The idea of consistency came from Laplace and informally may be defined as : Let \(\mathcal{P}\) be a set of probability measures on a sample space \(\mathcal{X}\), Î be a prior on \(\mathcal{P}\). The posterior is said to be consistent at a true value P 0 if the following holds: For sample sequences with P 0 probability 1, the posterior probability of any neighborhood U of P 0 converges to 1.
The choice of neighborhoods Udetermines the strength of consistency. One choice, when the sample space...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References and Further Reading
Antoniak CE (1974) Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann Stat 2:1152–1174
Barron A (1988) The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Technical Report 7, Department Statistics, University of Illinois, Champaign
Barron A, Schervish MJ, Wasserman L (1999) The consistency of posterior distributions in nonparametric problems. Ann Stat 27:536–561
Choi T, Ramamoorthi RV (2008) Remarks on consistency of posterior distributions. In: Pushing the limits of contemporary statistics: contributions in honor of Ghosh JK, vol 3 of Inst Math Stat Collect. Institute of Mathematical Statistics, Beachwood, pp 170–186
Coram M, Lalley SP (2006) Consistency of Bayes estimators of a binary regression function. Ann Stat 34:1233–1269
Diaconis P, Freedman D (1986) On the consistency of Bayes estimates. Ann Stat 14:1–67. With a discussion and a rejoinder by the authors
Ferguson TS (1974) Prior distributions on spaces of probability measures. Ann Stat 2:615–629
Ferguson TS (1983) Bayesian density estimation by mixtures of normal distributions. In: Recent advances in statistics. Academic, New York, pp 287–302
Freedman DA (1963) On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann Math Stat 34:1386–1403
Freedman DA (1965) On the asymptotic behavior of Bayes estimates in the discrete case II. Ann Math Stat 36:454–456
Ghosal S, Ghosh JK, van der Vaart AW (2000) Convergence rates of posterior distributions. Ann Stat 28:500–531
Ghosal S, van der Vaart AW (2001) Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann Stat 29:1233–1263
Ghosh JK, Ramamoorthi RV (2003) Bayesian nonparametrics. Springer Series in Statistics. Springer, New York
Kleijn BJK, van der Vaart AW (2006) Misspecification in infinite-dimensional Bayesian statistics. Ann Stat 34:837–877
Schwartz L (1965) On Bayes procedures. Z Wahrscheinlichkeitstheorie und Verw Gebiete 4:10–26
Shen X, Wasserman L (2001) Rates of convergence of posterior distributions. Ann Stat 29(3):687–714
Tokdar S, Chakrabarti A, Ghosh J (2010) Bayesian nonparametric goodness of fit tests. In: M-H Chen, DK Dey, P Mueller, D Sun, K Ye (eds) Frontiers of statistical decision making and Bayesian analysis. Inst Math Stat Collect. Institute of Mathematical Statistics, Beachwood
van der Vaart AW, van Zanten JH (2008) Reproducing kernel Hilbert spaces of Gaussian priors. In: Pushing the limits of contemporary statistics: contributions in honor of Ghosh JK, vol 3 of Inst Math Stat Collect. Institute of Mathematical Statistics, Beachwood, pp 200–222
van der Vaart AW, van Zanten JH (2009) Adaptive Bayesian estimation using a Gaussian random field with inverse gamma bandwidth. Ann Stat 37:2655–2675
Walker S (2004) New approaches to Bayesian consistency. Ann Stat 32:2028–2043
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Ghosh, J.K., Ramamoorthi, R.V. (2011). Posterior Consistency in Bayesian Nonparametrics. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_453
Download citation
DOI: https://doi.org/10.1007/978-3-642-04898-2_453
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04897-5
Online ISBN: 978-3-642-04898-2
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering