Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

By Dominique Bontemps. 2010

75 citations recorded by the source corpus.

This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises Theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and $C^{\alpha}$ classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5ff61d857a9454d1a62a0a239ed53dc80b4b3136

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Citations in the corpus

A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “Ghosal [20], Boucheron and Gassiat [8] and Bontemps [7].”
Large ball probabilities, Gaussian comparison and anti-concentration (2017, background): “∗ ∈ E G(r) = IP a+ξ ≤ r . Here ξ ∼ N(0,Σ). So, it suﬃces to compare two probabilities IP a+ξ ≤ r vs IP ξ G ≤ r for all r≥ 0. Existing results cover only very special cases; see e.g. Johnstone (2010); Bontemps (2011); Panov and Spokoiny (2015); Castillo (2012); Castillo and Nickl (2013); Belitser (20”
Bernstein - von Mises Theorem for growing parameter dimension (2013, background): “Similar conditions and results can be found in the literature for more specific models, see e.g. Bontemps (2011) for the Gaussian case.”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “We mention [4, 5, 14–16] as specific examples (see the discussion after Theorem 5.”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “For instance, one may investigate how the classical BvM theorem in finite-dimensions extends to parameter spaces of dimension that increases with n; see, [2, 15, 25] for results in this direction.”
BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS (2011, unknown): “26 92 v2 [ m at h. ST ] 2 3 Fe b 20 12 The Annals of Statistics 2011, Vol. 39, No. 5, 2626–2657 DOI: 10.1214/11-AOS920 c© Institute of Mathematical Statistics, 2011 BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS By B. T. Knapik1, A. W. van der Vaart and J. H. van Zanten”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “For the case of finite-dimensional posteriors with increasing dimension, see Ghosal [19] and Bontemps [4] for the case of regression or Boucheron and Gassiat [5] for discrete probability distributions.”
Dirichlet-Laplace priors for optimal shrinkage (2014, background): “[13] and [5] provided conditions on the prior for asymptotic normality of linear regression coefficients allowing the number of predictors p to increase with sample size n, with [13] requiring a very slow rate of growth and [5] assuming p ≤ n.”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “A number of BvM-type results have been obtained for the fixed finite-dimensional posterior with dimension increasing to infinity: Ghosal [16] and Bontemps [6] consider regression with a finite number of regressors, Ghosal [17] and Clarke and Ghosal [9] consider exponential families, and the case of ”