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Abstract:
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In recent years, shrinkage priors have received much attention in high-dimensional data analysis from Bayesian perspective. Compared with widely used spike-and-slab priors, shrinkage priors have better computational efficiency. But its theoretical properties, especially posterior contraction rate, which is important in quantifying estimate’s uncertainties, are not established in many cases. In this paper, we apply global-local shrinkage priors to high-dimensional multivariate linear regression with unknown covariance matrix. We show that when the prior is highly concentrated near zero and has heavy tail, the posterior contraction rates for both coefficients matrix and covariance matrix are nearly optimal. Our results hold when number of features p grows much faster than sample size n, which is of great interest in modern data analysis. For application, we show that a class of scale mixture of normal priors satisfies the conditions and can be easily implemented.
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