Posterior consistency in linear models under shrinkage priors

@article{Armagan2013PosteriorCI,
  title={Posterior consistency in linear models under shrinkage priors},
  author={Artin Armagan and David B. Dunson and Jaeyong Lee and Waheed Uz Zaman Bajwa and Nate Strawn},
  journal={Biometrika},
  year={2013},
  volume={100},
  pages={1011-1018}
}
We investigate the asymptotic behaviour of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution concentrates in neighbourhoods of the true parameter under simple sufficient conditions. These conditions hold under popular shrinkage priors given some sparsity assumptions. Copyright 2013, Oxford University Press. 
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References

SHOWING 1-10 OF 34 REFERENCES
Asymptotic normality of posterior distributions in high-dimensional linear models
We study consistency and asymptotic normality of posterior distributions of the regression coefficient in a linear model when the dimension of the parameter grows with increasing sample size. UnderExpand
Inference with normal-gamma prior distributions in regression problems
This paper considers the efiects of placing an absolutely continuous prior distribution on the regression coe-cients of a linear model. We show that the posterior expectation is a matrix-shrunkenExpand
Asymptotics for lasso-type estimators
We consider the asymptotic behavior of regression estimators that minimize the residual sum of squares plus a penalty proportional to Σ ∥β j ∥γ for some y > 0. These estimators include the Lasso as aExpand
GENERALIZED DOUBLE PARETO SHRINKAGE.
TLDR
The properties of the maximum a posteriori estimator are investigated, as sparse estimation plays an important role in many problems, connections with some well-established regularization procedures are revealed, and some asymptotic results are shown. Expand
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors
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 linearExpand
Generalized Beta Mixtures of Gaussians
TLDR
A new class of normal scale mixtures is proposed through a novel generalized beta distribution that encompasses many interesting priors as special cases and develops a class of variational Bayes approximations that will scale more efficiently to the types of truly massive data sets that are now encountered routinely. Expand
Bayesian lasso regression
The lasso estimate for linear regression corresponds to a posterior mode when independent, double-exponential prior distributions are placed on the regression coefficients. This paper introduces newExpand
Mixtures of g Priors for Bayesian Variable Selection
Zellner's g prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency issues. In this article we study mixtures of g priors as anExpand
The Bayesian Lasso
The Lasso estimate for linear regression parameters can be interpreted as a Bayesian posterior mode estimate when the regression parameters have independent Laplace (i.e., double-exponential) priors.Expand
Variational Bridge Regression
  • A. Armagan
  • Mathematics, Computer Science
  • AISTATS
  • 2009
TLDR
Results suggest that the proposed method yields an estimator that performs significantly better in sparse underlying setups than the existing state-of-the-art procedures in both n > p and p > n scenarios. Expand
...
1
2
3
4
...