High dimensional Bernstein-von Mises: simple examples
By Iain M. Johnstone. 2010
57 citations recorded by the source corpus.
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.
Subfields: Artificial Intelligence, Statistics and Probability
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In Artificial Intelligence
- Maxim Panov
- Vladimir Spokoiny
- Anirban Bhattacharya
- Peter J. Bickel
- Yun Yang
- Debdeep Pati
- B. Kleijn
- Tommaso Cai
In Statistics and Probability
- Maxim Panov
- Vladimir Spokoiny
- B. Kleijn
- Anirban Bhattacharya
- Peter J. Bickel
- Yun Yang
- Debdeep Pati
- Victor Chernozhukov
Citations in the corpus
- Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, methodology): “We mention Ghosal (1999, 2000), Boucheron and Gassiat (2009), Johnstone (2010) and Bontemps (2011) for specific examples.”
- Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “…(BvM) theorem under the total variation metric holds fairly generally and guarantees asymptotically nominal coverage of Bayesian credible intervals, the story of the frequentist validity of credible intervals/bands in infinite dimensional models is more complicated and delicate [16, 20, 24, 27].”
- The semiparametric Bernstein-von Mises theorem (2010, background): “a class of nonparametric priors based on infinite-dimensional exponential families. Boucheron and Gassiat [4] consider the Bernstein–von Mises theorem for families of discrete distributions. Johnstone [20] studies various marginal posteriors in the Gaussian sequence model. Notation and conventions. T”