High-Dimensional Multivariate Posterior Consistency Under Global-Local Shrinkage Priors
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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 × q coefficients matrix B. We show that our method can consistently estimate B even when p > n and even when p grows at nearly exponential rate with the sample size. This appears to be the first result of its kind and the first paper to study posterior consistency for the Bayesian multivariate linear regression model in the ultra high-dimensional setting. Simulations and data analysis show that our model has excellent finite sample performance.
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