Asymptotic normality of posterior distributions in high-dimensional linear models

By Subhashis Ghosal. 1999

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Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, methodology): “ space and the total available information in the data. Some recent results study the impact of a growing parameter dimension p n on the quality of Gaussian approximation of the posterior. We mention Ghosal (1999, 2000), Boucheron and Gassiat (2009), Johnstone (2010) for specic examples. See the dis”
Bayesian variable selection for high dimensional generalized linear models : Convergence rates of the fitted densities (2007, unknown): “34 58 v1 [ m at h. ST ] 1 8 O ct 2 00 7 The Annals of Statistics 2007, Vol. 35, No. 4, 1487–1511 DOI: 10.1214/009053607000000019 c© Institute of Mathematical Statistics, 2007 BAYESIAN VARIABLE SELECTION FOR HIGH DIMENSIONAL”
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”
Bernstein - von Mises Theorem for growing parameter dimension (2013, background): “ot of parameters and with limited sample size. There is a number of papers in this direction recently appeared. We mention Ghosal et al. (2000); Ghosal and van der Vaart (2007) for the i.i.d. models, Ghosal (1999), Ghosal (2000) for high dimensional linear models, Boucheron and Gassiat (2009), Kim (”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “• In Ghosal (1999) σn is constant, which does not allow the application to the Gaussian sequence model.”
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): “eau [32] for linear functionals of probability density functions. 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 ”
Asymptotic Normality of Posterior Distributions for Exponential Families when the Number of Parameters Tends to Infinity (2000, unknown): “Although priors constructed by Ghosal et al. [11] lead to a better rate of convergence of the posterior, computation of the posterior for those priors is much more involved.”
Dueling Posterior Sampling for Preference-Based Reinforcement Learning (2019, result): “We hypothesize that existing results on asymptotic consistency of Bayesian linear regression [19] and Gaussian process regression [34] could be leveraged toward extending notions of consistency and regret toward these credit assignment models.”
An Adaptive Sequential Monte Carlo Sampler (2010, background): “For example, standard Bayesian asymptotics (Ghosal 1999) suggest that if θ0 is the true parameter value and Σt is the posterior variance after t observations, then the posterior for Σ −1/2 t (θt− θ0) will converge to a standard Gaussian distribution.”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “A similar result about asymptotic normality of the posterior in a linear regression model can be found in [14].”
High dimensional Bernstein-von Mises: simple examples (2010, background): “Ghosh and Ramamoorthi (2003); Ghosal and van der Vaart, (2010)).”
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.”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “Ghosal [26], Boucheron and Gassiat [9] and Bontemps [8].”