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High-dimensional multivariate posterior consistency under global-local shrinkage priors. (English) Zbl 1403.62134

Summary: We consider sparse Bayesian estimation in the classical multivariate linear regression model with \(p\) regressors and \(q\) response variables. In univariate Bayesian linear regression with a single response \(y\), shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a \(p \times q\) coefficients matrix \(\mathbf{B}\). We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when \(p > n\) and even when \(p\) grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive \(\mathsf{R}\) package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62F15 Bayesian inference
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators

Software:

R2GUESS; R; MBSP; glmnet
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References:

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