×

The Bernstein-von Mises theorem under misspecification. (English) Zbl 1274.62203

Summary: We prove that the posterior distribution of a parameter in misspecified LAN parametric models can be approximated by a random normal distribution. We derive from this that Bayesian credible sets are not valid confidence sets if the model is misspecified. We obtain the result under conditions that are comparable to those in the well-specified situation: uniform testability against fixed alternatives and sufficient prior mass in neighbourhoods of the point of convergence. The rate of convergence is considered in detail, with special attention for the existence and construction of suitable test sequences. We also give a lemma to exclude testable model subsets which implies a misspecified version of Schwartz’ consistency theorem, establishing weak convergence of the posterior to a measure degenerate at the point at minimal Kullback-Leibler divergence with respect to the true distribution.

MSC:

62F15 Bayesian inference
62F25 Parametric tolerance and confidence regions
62F12 Asymptotic properties of parametric estimators
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] S. Bernstein, Theory of probability , Moskow (1917).
[2] R. Berk, Limiting behaviour of posterior distributions when the model is incorrect , Ann. Math. Statist. 37 (1966), 51-58. · Zbl 0151.23802
[3] R. Berk, Consistency of a posteriori , Ann. Math. Statist. 41 (1970), 894-906. · Zbl 0214.45703
[4] P. Bickel and J. Yahav, Some contributions to the asymptotic theory of Bayes solutions , Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 11 (1969), 257-276. · Zbl 0167.17706
[5] O. Bunke and X. Milhaud, Asymptotic behavior of Bayes estimates under possibly incorrect models , Ann. Statist. 26 (1998), 617-644. · Zbl 0929.62022
[6] A. Dawid, On the limiting normality of posterior distribution , Proc. Canad. Phil. Soc. B67 (1970), 625-633. · Zbl 0211.50802
[7] S. Ghosal, J. Ghosh and A. van der Vaart, Convergence rates of posterior distributions , Ann. Statist. 28 (2000), 500-531. · Zbl 1105.62315
[8] P. Huber, The behavior of maximum likelihood estimates under nonstandard conditions , Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1 , 221-233. University of California Press, Berkeley (1967). · Zbl 0212.21504
[9] B. Kleijn and A. van der Vaart, Misspecification in Infinite-Dimensional Bayesian Statistics. Ann. Statist. 34 (2006), 837-877. · Zbl 1095.62031
[10] P. Laplace, Théorie Analytique des Probabilités (3rd edition) , Courcier, Paris (1820).
[11] L. Le Cam, On some asymptotic properties of maximum-likelihood estimates and related Bayes estimates , University of California Publications in Statistics 1 (1953), 277-330.
[12] L. Le Cam and G. Yang, Asymptotics in Statistics: some basic concepts , Springer, New York (1990). · Zbl 0719.62003
[13] E.L. Lehmann., Theory of Point Estimation , John Wiley & Sons, New York. · Zbl 0870.62018
[14] R. Von Mises, Wahrscheinlichkeitsrechnung , Springer Verlag, Berlin (1931). · JFM 57.1499.04
[15] D. Pollard, Convergence of Stochastic Processes , Springer, New York (1984). · Zbl 0544.60045
[16] L. Schwartz, On Bayes procedures , Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 4 (1965), 10-26. · Zbl 0158.17606
[17] X. Shen and L. Wasserman, Rates of convergence of posterior distributions , Ann. Statist. 29 (2001), 687-714. · Zbl 1041.62022
[18] A. van der Vaart, Asymptotic Statistics , Cambridge University Press, Cambridge (1998). · Zbl 0910.62001
[19] A. Walker, On the asymptotic behaviour of posterior distributions , J. Roy. Statist. Soc. B31 (1969), 80-88. · Zbl 0176.48901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.