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August 2020 Bayesian linear regression for multivariate responses under group sparsity
Bo Ning, Seonghyun Jeong, Subhashis Ghosal
Bernoulli 26(3): 2353-2382 (August 2020). DOI: 10.3150/20-BEJ1198

Abstract

We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors; (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving the $\ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein–von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order $1/2$. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.

Citation

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Bo Ning. Seonghyun Jeong. Subhashis Ghosal. "Bayesian linear regression for multivariate responses under group sparsity." Bernoulli 26 (3) 2353 - 2382, August 2020. https://doi.org/10.3150/20-BEJ1198

Information

Received: 1 July 2018; Revised: 1 January 2020; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193963
MathSciNet: MR4091112
Digital Object Identifier: 10.3150/20-BEJ1198

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 3 • August 2020
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