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High dimensional Bernstein-von Mises: simple examples

Iain M. Johnstone

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In Gaussian sequence models with Gaussian priors, we develop some simple examples to illustrate three perspectives on matching of posterior and frequentist probabilities when the dimension p increases with sample size n: (i) convergence of joint posterior distributions, (ii) behavior of a non-linear functional: squared error loss, and (iii) estimation of linear functionals. The three settings are progressively less demanding in terms of conditions needed for validity of the Bernstein-von Mises theorem.

Chapter information

James O. Berger, T. Tony Cai and Iain M. Johnstone, eds., Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2010), 87-98

First available in Project Euclid: 26 October 2010

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Primary: 62E20: Asymptotic distribution theory
Secondary: 62F15: Bayesian inference

high dimensional inference Gaussian sequence linear functional squared error loss posterior distribution frequentist

Copyright © 2010, Institute of Mathematical Statistics


Johnstone, Iain M. High dimensional Bernstein-von Mises: simple examples. Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown, 87--98, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2010. doi:10.1214/10-IMSCOLL607.

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