The Annals of Statistics

Misspecification in infinite-dimensional Bayesian statistics

B. J. K. Kleijn and A. W. van der Vaart

Full-text: Open access


We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback–Leibler divergence with respect to P0. An entropy condition and a prior-mass condition determine the rate of convergence. The method is applied to several examples, with special interest for infinite-dimensional models. These include Gaussian mixtures, nonparametric regression and parametric models.

Article information

Ann. Statist., Volume 34, Number 2 (2006), 837-877.

First available in Project Euclid: 27 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G08: Nonparametric regression 62G20: Asymptotic properties 62F05: Asymptotic properties of tests 62F15: Bayesian inference

Misspecification infinite-dimensional model posterior distribution rate of convergence


Kleijn, B. J. K.; van der Vaart, A. W. Misspecification in infinite-dimensional Bayesian statistics. Ann. Statist. 34 (2006), no. 2, 837--877. doi:10.1214/009053606000000029.

Export citation